Download Computer simulation of the dynamics of aqueous solvation

Computer simulation of the dynamics of aqueous solvation
Mark Maroncelli and Graham R. Fleming
Citation: The Journal of Chemical Physics 89, 5044 (1988); doi: 10.1063/1.455649
View online: http://dx.doi.org/10.1063/1.455649
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Computer simulation of the dynamics of aqueous solvation
Mark Maroncelli
Department of Chemistry, The Pennsylvania State University, 152 Davey Laboratory, University Park,
Pennsylvania 16802
Graham R. Fleming
Department of Chemistry and the James Franck Institute, The University of Chicago, Chicago, Illinois
60637
(Received 7 June 1988; accepted 29 June 1988)
Equilibrium and nonequilibrium molecular dynamics computer simulations have been used to
study the time dependence of solvation in water. The systems investigated consisted of
monatomic ions immersed in large spherical clusters ofST2 water. Relaxation of the solvation
energy following step junction jumps in the solute's charge, dipole moment, and quadrupole
moment have been determined from equilibrium molecular dynamics (MD) simulations under
the assumption of a linear solvation response. The relaxation times observed differ
substantially depending on the type of mUltipole jump and the charge/size ratio ofihe solute.
These results could not be quantitatively understood on the basis either of continuum or
molecular theories of solvation dynamics currently available. Even the qualitative picture of a
distribution of relaxation times which monatonically increases with distance away from the
solute is not correct for the systems studied. This lack of agreement is partially explained in
terms of the structured environment of the first solvation shell of aqueous solutes. However,
translational mechanisms of polarization decay and effects due to the finite distribution of
charge within solvent molecules, which should be operative in less structured solvents as well,
also contribute to deviations from theoretical predictions. The validity of a linear response
approach has been examined for the case of charge jumps using nonequilibrium simulations.
The observed dynamics are not generally independent of the size of the charge jump and thus
linear response theories are not strictly applicable. In most cases, however, predictions based
on a linear response calculation using the equilibrium dynamics of the appropriate reference
system still provide a reasonable description of the actual nonequilibrium dynamics.
I. INTRODUCTION
Solvation and solvent effects are of major importance in
many areas of chemistry and their study has long enjoyed a
venerable place in chemical research. 1 A question that has
only lately come to the forefront in this area is "How fast is
solvation?" That is, how long does it take a polar solvent to
respond when the electronic makeup of a solute changes?
Driving the current interest in this question are recent theories that point to the importance of dynamical solvent effects
on charge transfer reactions in polar solution. 2- 7 The
strength of the coupling between a reacting system and its
solvent surroundings depends sensitively on the intrinsic
rate of reaction compared to the rate at which the solvent can
act to stabilize the attendant charge redistributions. While
this connection is theoretically well established, our understanding of the dynamical aspects of solvation is far from
complete. In the present work we report results of molecular
dynamics computer simulations aimed at learning more
about this fundamental problem.
The original theoretical treatments describing solvation
of newly formed ions or dipoles modeled the solvent in terms
of a homogeneous dielectric continuum, completely specified by its experimental dielectric dispersion E(W).8-12 As in
nearly all current models, the solute was viewed as a spherical cavity containing a point charge, point dipole, etc. 13 For
the simplest case of a Debye-type E( W ), such continuum
models predict an exponential relaxation of the solvation
energy in response to a change in the solute charge, dipole
moment, etc. The response time is approximately equal to
the solvent's longitudinal relaxation time T L' 14 In polar solvents, the longitudinal relaxation time is typically much
shorter than time scales for single-solvent reorientation, reflecting the fact that the solvation response involves the coupled response of many molecules. This T L prediction forms
an important benchmark for comparison of experimental
results and more realistic theoretical models.
Solvation times are measured experimentally by following the time-dependent shift ofthe fluorescence spectrum of
a dipolar probe solute after ultrafast excitation. Recently,
several groups have made such measurements on a wide
range of solute/solvent combinations. 15-20 A number of general observations can be made based on the results to date.
Solvation times are often close to, but nearly always longer
than T L' In solvents with very high dielectric constants, observed times can be more than an order of magnitude l6
greater than the T L prediction. An approximately linear correlation appears to exist between the ratio of observed solvation time to TL and solvent dielectric constant. 2 1 Finally, the
solvation energy typically decays nonexponentially in
time. 16 The experimental data thus deviate in several ways
from predictions of a homogeneous continuum models.
These deviations can most reasonably be assigned to the in-
© 1988 American Institute of Physics
0021-9606/88/205044-26$02.10
5044
J. Chern. Phys. 89 (8),15 October 1988
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M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
fluence of molecular aspects of solvation, not accounted for
in the simple continuum picture.
A number of more sophisticated theories hav.e recently
been proposed that attempt to go beyond a simple continuum description. Inhomogeneous continuum models 22 as
well as treatments that explicitly consider the molecular nature of the solvent23- 28 have been reported. In general, these
models all predict deviations from simple continuum behavior qualitatively like those observed in experiment. The most
applicable of the molecular models, the dynamical MSA
model,26-28 is even able to semiquantitatively reproduce
many experimental observations. 21 The insight provided by
these studies into the causes for failure of the simple continuum description can be loosely paraphrased as follows: Near
to the solute there are insufficient numbers of molecules to
attain the full cooperativity required of the 7"L response.
Here the response is slower, more single-particle-like than
7"L' Solvent far away, however, does look to the solute like a
continuum fluid, and there 7"L pertains. In fact, there is actually a continuous distribution of solvent relaxation times as a
function of distance from the solute that ranges between
slow, single particle times up to 7"L' Since the net relaxation is
a superposition of all of these responses, it is both nonexponential and slower than 7"L' The correlation between the deviation of the actual solvation time from 7"L and the solvent
dielectric constant can also be rationalized on this basis.21.22
Although substantial experimental and theoretical
progress has been made in the last few years, our understanding of the dynamics of solvation is far from complete. More
experimental data, especially in high dielectric constant solvents, is needed to verify the generality in the trends discussed above. On the theoretical side, molecular theories
have yet to advance to the stage where quantitative agreement with observed behavior has been achieved. More importantly, there are a number of simplifying assumptions
common to current theories that need to be explored before
we can be confident that even the correct qualitative picture
has been reached. Two of the most important are that the
solvent behaves linearly with respect to changes in the solute, and that only solvent reorientational mechanisms, i.e.,
those detected in E( W ), are important in the solvation response.
The research reported here is an attempt to provide new
data with which to test some ofthese assumptions. Our "experiments" are molecular dynamics (MD) computer simulations. The molecular level detail afforded by this type of
study is vastly superior to that obtained from laboratory experiments and is well suited to investigating the soundness of
current models. In this initial study we have focused on simulations of idealized probe solutes in a fairly realistic model
solvent. The solutes studied are spherical ions of varying size
and charge-the same type of solute considered by theory,
but rather far from the complex probe molecules used experimentally. The solvent used is the ST2 model of water. 29
This model provides an accurate representation of many of
the properties of liquid wate~9.30 and is much closer to an
experimental solvent than are the sorts of models employed
in theoretical studies.
The choice of water as a solvent was made primarily on
5045
the basis of the vast amount already known about aqueous
solvation. 31 Especially in terms of computer simulations, the
number of studies of aqueous solvation is at least an order of
magnitude greater than for all other solvents combined. Traditionally, simulations have mainly concerned the equilibrium structures and energetics of solvation in water. Solutes
ranging from simple atomic32 and ionic33 species, to complex
polyatomic molecules,34,35 and even the quantum mechanical electron solvation36,37 in water have been studied in this
context. A number of simulations have also addressed dynamical aspects of aqueous solvation, For example, how
reorientation and diffusion times of water molecules are perturbed by solutes,35,38-40 the mechanism of diffusion of simple ions,41-43 and the dynamical effect of water on an SN2
reaction4 have all been studied. While such studies relate to
the quantity of interest here, namely the rate at which the
solvation energy relaxes after a change in the solute charge
distribution, none provides this sort ofinformation directly,
Only one study, that of Engstrom et 01.44 concerning quadrupolar relaxation of ions, actually provides solvation rates
that can be directly compared with our results. Two further
studies have, however, observed other aspects of the response to nonequilibrium changes in an aqueous solute and
are closely related to our work, In an early simulation Rao
and Berne45 studied the time scale for structural relaxation
after the ionization process Ne_Ne2+. Very recently,
Karim et 01.46 have studied the polarization response to
changes of the dipole moment of a large spherical solute.
Results of these latter studies will also be discussed later.
The format of the remainder of the paper is as follows:
Methods used in performing both equilibrium and nonequilibrium MD simulations are described in Sec. II. This section
begins by discussing the linear response connection between
the time dependence of the solvation process and time correlation functions (TCFs) of appropriate quantities observed
in an equilibrium simulation. Details of the solutes studied
and simulation algorithms are also provided here. For reasons discussed in Sec. II, we have chosen to carry out simulations in large spherical clusters of water rather than using the
more typical periodic boundary conditions. Section III summarizes some of the properties ofthese clusters. We demonstrate that the environment seen by a solute at the center of
such a cluster is virtually indistinguishable from that of bulk
ST2 solvent at the experimental density. In Sec. IV we begin
the discussion of solvation by examining static aspects of the
hydration structures and solvation energies observed for
various solutes studied. Although such equilibrium results
are not new, they form an important background for understanding the differences in dynamics exhibited by the different solutes. Section V concerns dynamics observed in the
equilibrium simulations. How the solutes influence the dynamics of surrounding waters as well as the TCFs related to
solvation are presented. In Sec. VI we compare the timedependent solvation responses obtained in Sec. V to the predictions of the simple continuum model and the dynamical
MSA model mentioned above. Neither of these theories provide a quantitative account of the observed behavior and the
reasons for this failure are discussed in terms of molecular
mechanisms of relaxation. Section VII details the results of
J. Chem. Phys., Vol. 89, No. 8,15 October 1988
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M. Maroncelli and G. A. Fleming: Dynamics of aqueous solvation
5046
nonequilibrium simulations in which the charge on a solute
was jumped and subsequent relaxation observed. Comparison between the observed nonequilibrium response and the
response derived from the equilibrium simulations under the
linear response assumptions allows us to determine the limits of validity of the linear response approximation. Section
VIII concludes with a summary of our main results and a
discussion of how these results relate to current ideas about
solvation dynamics. Finally, in the Appendix we describe
our preliminary determination of the dielectric properties of
ST2 water.
TABLE I. Summary of simulations.
SimNo.
II. METHODS
The simulations we have used for studying the dynamics
of solvation are of two types. The first is a standard equilibrium MD simulation in which the solute properties are independent of time. From such simulations we obtain information concerning the equilibrium structure and dynamics that
exist about specific solutes. Equilibrium simulations also allow for calculation of the solvation response that would result from small variations of the solute's charge distribution.
The fluctuation-dissipation theorem connects fluctuations
of the electrical potential produced by solvent motions to the
dynamic response of the solvent to changes in the solute's
electrical properties. We will consider only the linear response regime and thus the eqUilibrium simulations provide
accurate predictions of the solvation response to "small" solute changes. What is meant by a small change can be addressed empirically through comparison with the second
type of simulation. In these nonequilibrium simulations we
take a preequilibrated sample and at some moment instantaneously alter the solute's charge. We then observe the subsequent evolution of the system and obtain the solvation response directly in time. A summary of all of the simulations
is contained in Table I.
Before describing the details of the simulations we first
discuss the formalism used to relate nonequilibrium solvation dynamics to the time correlation functions obtained
from equilibrium MD. We begin with the general problem of
a Hamiltonian which can be broken up into an unperturbed
part H (0) and a perturbation H' as
H=H(O)+H',
Equilibrium simulations·
299
512
291
512
1
2
3
4
5
6
7
8
9
10
Temp.(K)
No.mols
Solute
LO
LO
L+
L+
L+
L-
496
236
496
495
240
236
507
252
252
123
59
251
252
252
12
13
14
15
16
17
18
19
20
21
22
23
Nonequilibrium simulationsb
240
240
240
240
252
SO-S +!
252
SO-S+
252
S+-SO
11
Description
LO:
L +:
L-:
L +!:
large neutral
large cation
large anion
large partial
cation
small neutral
small cation
small partial
cation
S+:
S +!:
dt = 1 fs
n = 37
n=36
n=4O
n=4O
n=4O
n=4O
n=27
LO-L +
L +!-L +
L +-LO
LO-L-
Symbol
SO:
250 PS,
dt= 5 fs
300
302
300
299
297
296
294
295
297
292
298
292
299
299
SO
SO
SO
SO
SO
S +!
S+
S+
Solute designations
if
2.25
2.25
2.25
2.25
1.00
1.00
1.00
Notes
E'
q(a.u.)
70.0
70.0
70.0
70.0
0.00
+1.00
-1.00
+0.50
1.00
1.00
1.00
0.00
+1.00
+0.4444
• All equilibrium simulations were of 50 ps duration using a time step of 2 fs
unless otherwise noted.
bSimulations numbers 17-23 are nonequilibrium simulations in which the
charge was jumped in the direction described by L O-L +. etc. n here
designates the number of jumps performed.
C Solute-solute U parameters uand Eare given in units of the U parameters
oftheST2 water model (u= 3.IOA. E = 0.075 75 kcal/mol). The solutesolvent LJ interactions were calculated from these using the usual Lorentz-Berthelot combining rules.
where
H'=IX;Fj(t).
(2.1)
j
The perturbation terms X; Fj (t) couple the system variables
X; to the time-dependent perturbation F(t). In our application, the Fj (t) represent the time varying solute charge distribution and the X; are components of the electric field produced by the solvent that act on the solute. The first order
change in the system due to the perturbation is given by the
well-known result47 ;
(X;(t»(l)
=~
J
It
-
Fj(t')¢lij{t- t')dt'
(2.2)
'"
The pulse response function ¢I ij (t) describes the linear response of the ensemble-averaged (denoted by ( » system
property Xi to a delta function Fj perturbation. (The superscript 1 on (X; (t» denotes that this is the change to first
order in F.) Equation (2.3) relates this pulse response function to a time correlation function of the system variables in
the absence of the perturbation. In this expression X represents the time derivative of X and the ( ) (0) denotes an ensemble average calculated with H = H(O). We wiII be concerned with the response to a step function change in solute
properties, so we consider F( t) of the form
O,
with
¢lij(t)
= _1_ (X;X;(t»(O).
kT
(2.3)
= {Fj'
t<O
r;;. 0'
For such an F(t), Eq. (2.2) can be integrated to yield
Fj(t)
(2.4)
J. Chern. Phys., Vol. 89, No.8, 15 October 1988
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5047
M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
(Xi (t) ) (1)
-
(Xi ( 00
»(\)
=_1 "F [(XX(t»(O)
kT"'7 J
I
J
- (Xi)(O)(Xj
(2.5)
) (0) ] ,
where (Xi ( 00 » denotes the value of Xi when the system has
equilibrated to the perturbation. Equation (2.5) is the general fluctuation-dissipation expression we will use to obtain
the solvation response from our equilibrium simulations.
We now specialize the above general result so as to consider the solvation response to changes in the first three multipole moments of a solute's charge distribution. In particular, we will examine the relaxation of the solvation energy
after step function changes in either a point charge, dipole,
or axial quadrupole located at the center of the solute. The
electrostatic free energy of solvation of such point multipoles
can be written
E so1v (t)
=
{
1I2q(t)( V(t»
charge
- 1I2f.lz (t) (Ez (t» dipole
- 1I2Qzz (t) ( Vzz (t) )
,
quadrupole
(2.6)
where q is the magnitude of the charge and f.lz and Qzz are the
magnitudes of the dipole and quadrupole moments which
are assumed to lie along the z direction. The quantities V, E z '
and Vzz are the appropriate electric potential, field, and field
gradient ( - a 2 V / azZ) components at the site of the multipole induced by its interaction with the solvent. Similar expressions, without the factors of 112 and the ensemble averaging, also provide the perturbation terms in the
Hamiltonian, Eq. (2.5), with changes in q, f.l, and Q being
the F terms and the reaction fields V, E z , and Vzz being the
system properties X. For simplicity we will consider only the
effect of changes in magnitude of the dipole and quadrupole
rather than allowing the direction to change as well. (Under
this restriction the i-j cross correlations that would otherwise arise between different field components do not contribute.)
Let us consider the quadrupole case as an example. The
time-dependent perturbation corresponding to a step function change in quadrupole moment of magnitude 6.Qzz is
(2.7)
and the response to the solvation free energy 6.Eso1v (t) to
such a change is given to first order by
6.E ~I~ (t) - 6.E !~I~ ( 00 )
= -
~ Qzz [ ( Vzz (t) ) (1) -
(Vzz (
- 2lT Qzz6.Qzz [ (Vzz Vzz (t»
00 ) ) (I) ]
(0)
- (Vzz)(O)(Vzz)(O)]
1
- 2kT Qzz6.Qzz (8Vzz 8Vzz (t».
(2.8)
In these expressions Qzz denotes the total magnitude of the
quadrupole moment after the 6.Qzz change and 8Vzz is the
fluctuation in the zz field gradient component. The superscripts (1) and (0) again designate that the energies are
correct to first order in H' and that the ensemble averaging of
the Vzz quantities is performed in the absence of H'. We will
hereafter omit these superscripts.
Equations analogous to Eq. (2.8) for the charge and
dipole cases are obtained by substituting the appropriate
quantities for Vzz and Qzz. For discussing the solvation time
dependence it is usually more convenient to work with the
normalized response function Set) defined by
S(t)
=
6.Eso1v (t) - 6.Eso1v (
00 )
6.Eso1v (0) - 6.Eso1v (
00 )
(2.9)
rather than with the energies themselves. Within the linear
response approximation, the S( t) response functions appropriate to step changes in the three point multipoles are equal
to three normalized time correlation functions [C( t)] as
follows:
Equilibrium TCF
Nonequilibrium
response
(8V8V(t»
(8V2)
(8Ez8Ez (t»
(8E;)
6.q
C(t) -Set)
(2.10)
6.p,z
(8Vzz 8Vzz (t»
(8V~)
6.Qzz
These equations represent the desired final result connecting
the temporal decay of fluctuations in the quantities V, E z ,
and Vzz that we collect from the equilibrium MD simulations to the nonequilibrium solvation response S( t).
All of the simulations reported here involve spherical
solutes fixed at the center of spherical clusters of water solvent. The use of clusters for these studies has two advantages
over the more usual periodic boundary conditions. First,
maintaining spherical symmetry is helpful in that it allows
for the maximum possible averaging of observables since all
properties have only radial dependence. Thus, e.g., Vzz in
Eq. (2.9) isequivalentto Vxx and VYY ' sothatthecorrelation
functions we report for (8 Vzz 8Vzz (t» are actually averaged
over the independent x, y, and z directions. Second, by avoiding periodic boundary conditions we avoid having to make
approximate corrections for the long-ranged interactions
across cell boundaries. It has only been after many years of
debate that the use of reaction field 48 or Ewald summation49
techniques for treating periodic boundary conditions in polar solvents now seems to be well understood. 50 With clusters we bypass the complications involved with these techniques and work with a simple, well defined model system
whose simulation involves no such approximations. The
main drawback of using clusters is of course that the simulated system incorporates a large fraction of molecules near to
the surface of the cluster that have properties different from
those in the bulk. In order to study solvation in the liquid
phase we must work with clusters sufficiently large that edge
effects do not significantly affect the results. 5 I As we will
discuss in Sec. III we have found clusters containing 256512 solvent molecules to be adequate for the solvation properties of interest here.
J. Chern. Phys., Vol. 89, No.8, 15 October 1988
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5048
M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
The water-water potential chosen in this work is the
ST2 model of Stillinger and Rahman. 29 A water molecule in
this model consists of a single Lennard-Jones center at the
oxygen position (0'= 3.1 A and E= 0.076 kcallmol) and
four partial charges of magnitUde 0.24 e tetrahedrally arranged about this site that represent the two hydrogens and
two oxygen lone pairs. The hydrogen positions are 1 A from
the oxygen whereas the negative charges are at a distance of
0.8 A, producing a slight charge asymmetry in the molecule.
Solutes are constrained to the center of the cluster and interacted with the water solvent via a (6-12) Lennard-Jones
term plus Coulomb terms between the water charges and a
centered point charge. Details of the solute parameters are
provided in Table I. No interaction cutoffs were employed in
this work so that all water-water and water-solute interactions were explicitly included in the calculations. The cluster
was maintained using a spherical confining potential that
reflected waters attempting to evaporate from the cluster.
Even at 298 K water molecules are sufficiently strongly
bound that few molecules attempt to leave the cluster during
a 50 ps simulation. The placement of the confining potential
was chosen such that its presence had no observable effect on
the cluster properties.
Integration of the equations of motion was performed
using the Verlet algorithm 52 and the rigid body constraints
of the ST2 waters were maintained with the SHAKE53 and
RATTLE54 methods. A time step of 2 fs was used in almost
all simulations (Table I lists the exceptions). This step size
provided energy conservation to better than 2% over the
course of 50 ps for pure ST2 clusters. Energy conservation
for the small charged solutes was somewhat poorer (due to
the lack of a switching function in the solvent-solute interactions) so that the kinetic energy had to be rescaled several
times during the course of some of these runs. Comparison of
runs No. 15 and No. 16 showed that the influence of time
step had little influence on the observed solvation dynamics.
The equilibrium MD simulations typically consisted of
as-to ps equilibrium period followed by a 50 ps production
run during which data were collected. Rather than saving all
solvent coordinates, which would require excessive storage
for the large samples used, most observables were calculated
during the course of the run. Exceptions were the electrical
properties used in evaluating the solvation dynamics, which
were continuously output. The nonequilibrium simulations
were performed by running two calculations in parallel. The
first was a simulation in equilibrium with respect to the initial state of the solute; this was used to obtain starting points
for the nonequilibrium simulations. The second, nonequilibrium series of simulations were run using initial conditions
chosen from the eqUilibrium run at 1 ps intervals. In the
nonequilibrium runs the charge of the solute was instantaneously changed and the dynamics in the presence of the
altered solute then followed for 1 ps. Typically, a series of 40
charge jumps were simulated to produce averaged results. In
both the equilibrium and nonequilibrium simulations, properties were examined as a function of distance from the solute by dividing the system up into a series of 3-5 shell regions. The first and last shells were defined so as to
respectively include only waters that were nearest neighbors
1.5
::J
.=.
1
.......,
;..
Ul
c:
~ 0.5
o
o
o
1
234
5
6
ria
FIG. I. Water oxygen radial density profile ofa pure 512 molecule cluster.
Units are relative to the ST2 U diameter u. The numbers 1-4 mark the shel\
regions used in col\ecting data.
of the solute and surface waters, respectively (as judged by
the solute/solvent radial distribution function). The other
shells evenly divided the remaining radial space.
III. PURE SOLVENT CLUSTERS
Before describing the solvation results we first discuss
the nature of the solvent environment provided by the water
clusters used in these studies. Our main emphasis will be to
show that clusters containing 256 or 512 solvent molecules
are in most respects indistinguishable from the neat liquid.
We also describe a few features unique to the surfaces of
these clusters.
Figure 1 shows the radial density profile of a cluster of
512 ST2 molecules at 298 K. Distances in this and remaining
figures are given in units of the LJ diameter of the ST2 model, 0' = 3.10 A. We have examined how properties vary as a
function of radial position by dividing the sample into the
four regions marked in the figure. With few exceptions the
properties of water molecules in shells 1-3 are identical but
differ significantly from waters in the outermost region 4.
The dashed line in Fig. 1 denotes the experimental density of
water at this temperature. As far as the density is concerned,
the region out to about 40' is bulk-like for a cluster of this
size. The density falloff in the surface region takes place in
approximately one molecular diameter and is well represented
by
a
function
of' the
form per)
= 0.5po{1 - tanh[2(r - ro)/A]} with A = 1.10'.55
In Table II a number of static and dynamic properties
calculated from a simulation of a 512 molecule cluster are
compared to results obtained by others for the ST2 model in
periodic boundary simulations, and to experimental data for
water at 298 K. The density of the interior of the cluster is
- 1% low of the experimental value, which is considerably
closer agreement than is achieved in periodic simulations at
1 atm. As pointed out by Townsend et al. 55 this fortuitous
improvement seen in cluster simulations is due to the pressure exerted by the curved surface of the cluster. The poten-
J. Chern. Phys., Vol. 89, No.8, 15 October 1988
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M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
5049
TABLE II. Pure cluster properties (298 K).
Cluster N = 512"
Property
Surface
Interior
-7.2
6.9
6.0
2.2
0.988
-10.2
7.8
7.2
2.6
1.8
Density (gcm-')
- V / N(kcal/mol)d
TIde Tcxp
(r)
T2H e: 'T"exp
(r)
1.1
Periodicb
simulation
Experiment
0.925°
10.4'
_5'
8.2 (281 K)i
2.41
1.7i
0.997°
9.9°
-6&
2.5 k
• "Surface" waters were taken to be all molecules lying outside of a radius of r = 4.4u. At this radius, the density
has decreased to - 90% of the bulk density. All other waters were considered to be "interior" waters.
b The density value listed here is from a 1 atm constant-pressure simulation (c). The remaining values are from
constant volume simulations in which the density was constrained to be unity.
° Values from W. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey, and M. L. Klein, J. Chern. Phys.
79,926 (1983).
d V / N is the potential energy per molecule. For the clusters, division into surface and interior components was
done approximately, as described in the text.
"From R. A. Kuharski and P. J. Rossky, J. Chern. Phys. 82, 5164 (1985).
'Time constants of the I = I dipole and 1=2 H-H vector reorientational TCFs described in the text [Eqs.
(3.1) and (3.2»).
g Approximate value based on data from short simulations reported in A. Geiger, Ber. Bunsenges. Phys. Chern.
85,52 (1981); A. Geiger, A. Rahman, and F. H. Sti11inger, J. Chern. Phys. 70, 263 (1979).
h Approximate value based on a dielectric relaxation time of r D = 8.3 ps and using the approximate relation r d I
-{3EoI(2Eo + E~ )hD (see the text).
iD. A. Zichi and P. J. Rossky, J. Chern. Phys. 84, 2814 (1986).
i Gy. I. Szasz and K. Heinzinger, J. Chern. Phys. 49, 3467 (1983).
kNMR result ofJ. Jonas, T. DeFries, and D. J. Wilbur, J. Chern. Phys. 65, 582 (1976).
tial energy per water in the interior of the cluster we estimate
to be essentially identical to that observed in other simulations and to estimates of the experimental value. We did not
actually calculate the total potential energy per molecule in
each region but rather only the local bonding energy arising
from interactions between molecules separated by less than
1.50'. The bonding energies so calculated were the same for
shells 1-3. For surface waters the distribution oflocal bonding energies was shifted to more positive energies by about 3
kcal/mol, which is - 213 of the energy of a typical hydrogen
bond in the ST2 model (4.5 kcal/mol). The estimates ofthe
total potential energies listed in Table II were calculated
from the total cluster potential energy by assuming that this
difference in local bonding energies accounted for all of the
energetic difference between the two classes of molecules.
The dynamics of water molecules in the interior of the
cluster was also independent of position throughout regions
1-3 but considerably faster in region 4. In Fig. 2 we illustrate
the difference in dynamics using two reorientational time
correlation functions (TCFs). These correlation functions
are defined by
C/, (t)
=
(p/[u;'u;(t»),
(3.1)
where p/ denotes a Legendre polynomial of order I and u; is a
unit vector in the molecular frame. Here we have plotted the
correlation functions C 1d (t) and C2H (t) which involve the
unit vectors in the dipole (d) and H-H bond (H) directions.
These functions are not strictly exponential at long times due
to the exchange between interior and surface waters. (For
calculating these TCFs we assign molecules to regions only
once every 5 ps.) In Table II we compare characteristic time
constants of these TCFs to experimental and simulated val-
ues. The two time constants listed are rex' which is based on
an exponential fit of C(t) between 1-5 ps, a region that excludes the fast librational component ofthe motion, and the
average time (r),
1
00
(r)
=
C(t)dt
(3.2)
which does include this fast component. The reorientation
times we observe in the interior of the clusters are in good
agreement with times measured by others in bulk ST2 simulations. We note that our values for these quantities are more
o
___-1d
:;::; -1
u
......
C
.J
"-
.....
"
-2
-·'M,.
........
....-..
-.~
..............
0
1
2
3
Time
(ps)
4
5
FIG. 2. Reorientational time correlation functions ofthe water dipole (ld)
and H-H (2H) vectors in a pure (512) cluster. The first (l = 1; Id) and
second (/ = 2; 2H) order Legendre TCFs [Eq. (3.1)) are plotted for two
different regions in the cluster. Solid lines are for molecules in the bulk regions 1-3 and dotted lines correspond to surface waters (region 4) as defined in Fig. 1.
J. Chern. Phys., Vol. 89, No.8, 15 October 1988
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5050
M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
reliable than most available in the literature since the latter
are typically based on simulations of only a few picoseconds
duration. The times for bulk ST2 water are also in reasonable
accord with the dynamics of real water. The value of (1'2H) is
obtained from NMR measurements and is directly comparable to the simulated quantity. The experimental value of
1'ld -6 ps is only a rough estimate obtained using the relation due to Powles56 between this single particle time and the
dielectric relaxation time of 1'D = 8.3 pS.57 The ST2 model
results are within 30% of the experimental values for both of
these dynamical properties. Reorientation of surface waters
is faster than that of bulk water by a factor of 5%-40%
depending on which measure is used. It may seem somewhat
surprising that the surface waters are not much faster than
bulk waters, however, the small difference actually observed
is in keeping with the fact that roughly three hydrogen bonds
are still maintained at the surface.
One final aspect of the pure solvent clusters ofinterest is
the orientation of waters at the surface. Several studies have
previously shown that water5.58.59 and other polar molecules60 tend to orient such that their dipole direction lies
parallel to the cluster surface. This phenomenon is illustrated in Fig. 3(A) for a 512 molecule cluster. Here we show the
cosine distribution of the angle made between the dipole direction and the radius vector as a function of radial shell.
There is a strong preference for dipolar orientation perpendicular to the radial direction for molecules in the surface
layer (shell No.4, solid curve). This preference also propagates to the interior layers, although the orientation is small
for regions 1 and 2. These orientational correlations are one
of the few properties for which there is a difference observed
among interior shell regions Nos. 1-3. In addition to dipolar
(Al dipole
1.5
...,»
....
........
.c
.c
a
[ 0.5
'"
IV. SOLVATION STRUCTURE AND ENERGETICS
o
1.5
»
...,
....
.,
..........
.c
.c
a
:':';::::;~,
'"
d::
0.5
o
orientation there is a preferential orientation of the H-H
direction relative to the radius vector as shown in Fig. 3 (B) .
In this case the distinction between different intemallayers
is negligible and only the surface layer shows much of an
effect. The preferred arrangement of exterior waters is for
the dipole to point parallel to the cluster surface and for the
HOH plane to be perpendicular to this surface. Compared to
the bulk, a surface molecule with such an orientation has
chosen to give up one H atom for hydrogen bonding by projecting it out from the cluster rather than doing the same
with one of its negative charge sites (Q). The distinction
between Hand Q is due to the small asymmetry in the O-H
and O-Qbond lengths in the ST2 model, 1.0 compared to 0.8
A, respectively. We note that such preferential orientation
leads to a net positive surface charge in ST2 clusters as was
recently observed by Brodskaya and Rusanoy59 in much
smaller clusters.
We have centered the preceding discussion around pure
clusters of size N = 512. Essentially the same features are
also observed for the clusters of256 molecules which we also
employ in our solvation studies. In clusters of both sizes
there is a surface region of thickness -10' in which molecules are less strongly bound, have a preferential orientation,
and reorient slightly faster than in bulk water. The remainder of the cluster is indistinguishable from bulk water with
the exception of there being a slight residual preference for
molecules to align with their dipoles perpendicular to the
radial direction. This residual orientation does not noticeably affect the observed energetics or dynamics of interior
waters and should not influence our solvation results significantly. The primary difference between clusters of size 512
vs 256 is simply the numbers of molecules contained in the
surface and interior regions. For clusters of size 512 the
numbers are approximately N sun -162 (32%) and N int
- 350 (68%) and for 256 clusters they are Nsun - 106
( 41 %) and N int - 150 (59%). How the presence of surface
water influences the solvation properties depends of course
on the range of the relevant interactions considered and will
be discussed later.
~~~~~~~~~~-.~~~
-1
-0.5
0
0.5
1
COS (O*f)
FIG. 3. Angular distributions of water molecules in a 512 cluster as a function of position. The angular variable is cos ( u*r) where ris the vectorfrom
the cluster center to the oxygen position and u is a molecule fixed vector
specifying (A) the dipole direction, and (B) the H-H internuclear direction. (We define the dipole direction to point from 0 to the H-H bisector.)
Probabilities have been scaled such that a random distribution would correspond to P = 1. In panel (A) distributions for allfour regions of the cluster
(Fig. 1) are shown. The distributions of the H-H vector were virtually identical in regions 1-3 and in panel (B) we show only the average distribution.
We have performed equilibrium simulations with six solutesthatwedesignateLO,L + ,L - ,SO,S + ~,andS +
(see Table I). These solutes are of two sizes "L = large" and
"s = small." The small solutes have Lennard-Jones parameters identical to those of the ST2 oxygen center: O'ss = 3.1
A, and Ess = 0.076 kcallmol. The 0, +~, and + descriptors on the small solutes designate 0, + 0.4444, and + 1
units of atomic charge at the solute center. Large solutes all
have LJ parameters O'LL = 70 and ELL = 5.3 kcallmol. The
0, +, and - again denote 0, + 1, and - 1 charge. The
rationale behind the choice of U parameters for the large
solutes was to simulate a system of roughly the volume and
polarizability of the probe molecules employed in fluorescence Stokes shift measurements. The small solutes were
used to examine size effects on solvation dynamics and are
intended more to represent small polar groups in a large
molecule rather than to closely resemble common ions. In all
simulations the solutes were immobilized at the center of the
J. Chern. Phys., Vol. 89, No.8, 15 October 1988
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5051
M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
LO
2
o
2
o
+---~------~~~--------~~
2
+-------------7----------.~~
L+
2
o
o
2
f-------------1*~--------~~
,t'...
2
,/ \
,,i
-
I
.
- - - - - - - -~ - - - - -',--
o
o
o
4
2
0
2
\ ....
+------.--~--~----~~~~~
o
-1
4
..-,,/
-1
0
riC)
FIG. 4. Summary of solute-water radial distribution functions for all solutes studied (256 clusters). The solid curves are the Sol-O distribution and
thedottedcurvesareeithertheSol-Q(LO,L + ,SO,S + ,S +!) or Sol-H
distributions (L - ) where 0, Q, and H denote the water oxygen, negative
charge, and hydrogen sites, respectively.
solvent cluster. This constraint is consistent with the approach of most theoretical studies of solvation dynamics
and, based on previous simulations of ion motion,43 should
have negligible effect on the dynamics of interest.
Figures 4 and 5 and Table III document some features of
the solvation structure that exists about these solutes. These
data are from simulations of 256 molecule clusters (see Table III), however, results are unchanged in larger clusters.
Figure 4 shows the radial distribution functions (RDFs) of
the distances between the solute ("Sol") and two interaction
sites of the ST2 solvent. In all but the L case, the water sites
displayed are the oxygen atom (0) and negative charge (Q)
FIG. 5. Summary of the angular distributions of water molecules in the first
solvation shells of the 6 solutes studied (256 clusters). The angular variable
is cos ( fl·;') where r is the vector from the solute center to the water oxygen
position and u is a water-fixed vector. Solid lines are for u = the dipole direction and dashed lines are for u = the O-H bond direction (pointing from
to H). Probabilities have been scaled such that a random distribution
would correspond to P = I.
°
positions. For the L solute the water hydrogen atoms (H)
are shown instead of the negative charge sites. The Sol-O
RD Fs are similar for all of the large solutes and for S o. There
is a fairly broad but clearly defined first shell region and, in
addition, a small maximum denoting a second neighbor
shell. The small charged solutes have a much sharper first
neighbor peak (note vertical scale difference for S + ) in
their Sol-O RDF which is also closer to the solute by 0.10.2u. Solvation numbers, determined from areas under the
first peak, range from 7 for S + ~ and S + , to 16 for SO, to
-40 for all of the large solutes (Table III). The distribution
TABLE III. First shell solvent properties.
Solute
LO
L+
LSO
S +!
S+
No. nn"
42
40
43
16
7
7
O-solute radial distributionb
r(1- ) p(l- ) r(2 +)
r(1 +) p(1 +)
1.80
1.75
1.75
1.05
0.90
0.83
2.20
1.87
1.73
1.72
2.10
5.16
2.28
2.28
2.33
2.61
1.2
1.12
0.6
0.7
0.7
0.7
1.0
0.22
2.6
2.26
2.6
1.9
1.5
(bulk:
Water-water energetics"
E bind
No. Nhb
~Eb;nd
16.5
15.5
15.5
16.5
12.5
3.0
16.0
9
10
10
9
II
II
10
3.7
3.5
3.5
3.7
2.8
0.7
3.6)
• Number of water molecules in the first solvation shell as measured by the integral under the solute-oxygen
RDF up to the first minimum [r(1- )].
b Characteristics of the solute-oxygen radial distribution functions. r designates a distance from the solute in
units of the ST2u( 3.10 A) and p is the solvent oxygen density in units of molecules/ul [the average density is
essentially unity so that these values are equivalent to values of g( r)]. The designations I +, I - , and 2 +
refer, respectively, to the first maximum and minimum and second maximum of the RDFs.
"The binding energy (kcal/mol) of a molecule is defined here as the potential energy of interaction of the
molecule with all neighboring water molecules within a distance of 1.5u. The values Emnd and ~Emnd refer to
the energy at the maximum and FWHM of the distribution of binding energies of first shell waters. N hb is the
number of hydrogen bonds calculated from Emnd using an average H-bond energy of 4.5 kcallmol from pure
water simulations.
J. Chern. Phys., Vol. 89, No.8, 15 October 1988
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M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
5052
TABLE IV. Solute-solvent energies.
Solute
Nso''tl
Sim No.·
E,w (Tot)b
E,w(vdW)b
LO
LO
L+
L+
L-
496
236
496
240
236
507
252
123
59
251
252
3
4
5&6
7
8
9
10&11
12
13
14
15&16
- 32.9
-30.9
-96.2
- 88.7
- 89.7
-0.36
-0.44
-0.43
-0.21
- 32.7
- 179.0
- 32.9
- 30.9
-29.9
-28.5
-28.0
-0.36
-0.44
-0.43
-0.21
+ 1.8
+ 13.2
SO
SO
SO
SO
S +!
S+
E,w(El)b
%E(Shelll )C
-66.3
-60.2
- 61.7
57
62
57
- 34.5
-192.0
64
66
• These data are separated according to solute and cluster size (N",'v), In cases where more than one simulation
is listed the values shown are averages.
bPotential energies of solute-water interactions Esw in kcal/mol. The energy is decomposed into contributions
due to van der Waals (vdW) and electrostatic (E I) terms in the solvent-water interaction.
C %E(shell I) is the percentage of E,w (E I) due to first solvation shell waters.
functions for the charge sites differ in the expected manner
between the charged and uncharged species. Thus for L 0 and
SO, the Sol-Q and Sol-H RDFs are indistinguishable,
whereas for the charged solutes there is an excess of oppositely charged sites close to the solute.
Figure 5 shows the angular distributions of solvent molecules in the first solvation shell. The two angles depicted are
those between the Sol-O radius vector and the dipole direction (solid), and the radius vector and the O-H bond direction (dashed curves). The solvation structures that can be
deduced from these angular distributions and the radial distribution functions are well known from previous water simulations.32.33.38 All of the large solutes, as well as SO, exhibit
characteristics of "hydrophobic" solvation. Waters in the
first solvation shell tend to form a clathrate-like structure
around the solute, pointing three of their four charge sites
tangentially to the solute surface and the fourth radially
outward. By so doing virtually the same amount of hydrogen
bonding that is present in bulk water can be preserved.
(Quantitative comparisons between the water-water binding energy distributions of first shell solvent are provided in
Table III). There are some differences between the angular
distributions of the L solutes depending on charge but these
represent a modest perturbation on the dominant hydrophobic theme. Thus the dipole distributions of L + and L - are
skewed so as to bias the charge distribution slightly in the
appropriate directions. Based on the areas under the small-r
.shoulders in the Sol-Q and Sol-H RDFs of L + and L - ,
however, we estimate that fewer than 10% of the solvent
molecules actually point a charge directly at the solute.
(Note also that the dipole distributions of the uncharged
solutes are already slightly asymmetric such that there is a
small negative potential near the solute.) Solvation of the
S + solute represents the opposite extreme from that of the
L solutes, namely hydrophilic hydration. The structure here
is one in which first shell waters are very strongly oriented so
as to point one of their negative charge sites directly at the
solute (Fig. 5, again note the vertical scale difference). First
shell waters give up favorable interactions with one another
in order to pack as closely to the solute as possible and thereby maximize the solute-solvent attraction. As can be seen
from Table III, less than 1/5 of the water-water binding
energy of the bulk is preserved in the first solvation shell of
S + . The final solute, the partially charged S + !, is a case
intermediate between the hydrophobic and hydrophilic extremes. All of the features of its solvation structure appear to
reflect a superposition of those of solutes SO and S + , in
roughly comparable amounts. This superposition is evident
in the radial distribution functions, the angular distributions, and in the water-water energy distributions.
The energetics of solvation are considered in Table IV
and Fig. 6. We have not attempted to calculate solvation free
energies and so we compare only relative solute-water interaction energies, Esw. Several points can be made on the basis
of these energies. First, the total Esw is somewhat dependent
on the size of the cluster. This is to be expected for ionic
solutes. The electrostatic energy of interaction between a
charge and the polarization it induces in the solvent a distance r away varies as r- 3 for large r. Modeling a cluster as a
200
u
o
o
100
.--i
co
U
3:
o
III
W
<l
-100
-200
-200
-100
o
~Esw
100
200
Obs .
FIG. 6. Observed vs calculated solvation energy differences (boE. w ' kcall
mol) between solutes with different charges. The observed values were calculated from the E,,,, data in Table IV and the calculated values were obtained using the linear response result [Eq. (4.1) 1 (see the text).
J. Chern. Phys., Vol. 89, No.8, 15 October 1988
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M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
TABLE V. Continuum estimates of cluster size effects."
512
Cluster size
Percentage missing
128
256
64
V
E.
Vzz
10
0.1
0.001
13
0.2
0.003
Ssolute
16
0.4
0.01
21
0.9
0.04
V
E.
Vzz
23
1.2
0.06
29
2.4
0.2
Lsolute
36
4.8
0.6
47
10
2.3
Property
• Values listed are the percentage of the total electrostatic solvation energy
of an infinite sample that is missing when a finite cluster is used. These
numbers are based on treating the solvent as a continuum as discussed in
the text. V, E .. and Vzz refer to the energies of solvation ofa point charge,
dipole, and quadrupole, respectively.
homogeneous dielectric sphere, the fraction of the bulk solvation energy of an ion of radius ro in a cluster of radius R is
given by 1 - (rolR). The fractions of the bulk electrostatic
solvation energy of an L size ion expected for 512 and 256
molecule clusters are thus calculated to be 77% and 71 %,
respectively. The analogous fractions for S size solutes are
90% and 87%. (See also Table V.) The observed ratioofEsw
for L + in the 512 and 256 clusters is close to what is expected on this basis. Table IV also lists the part of Esw due to van
der Waals interactions. In the following sections we will be
interested in the time dependence of the solvation energy in
response to a change in solute charge. Comparing changes in
the total Esw to changes in the van der Waals component one
sees, as expected, the major effect is in the electrostatic component of Esw. The van der Waals energy also changes as a
result of changes in the first shell structure, however, this is a
secondary effect and amounts to less than 7% of the total
energy change in all cases. In the remaining analysis we
therefore concentrate exclusively on the energetics of the
electrostatic interactions and ignore the second order
changes in the van der Waals energies.
It is also instructive to examine what fraction of the
electrostatic part of Esw comes from interactions with the
first solvation shell. As shown in Table IV, even for the large
solutes, over half of the energy comes from nearest neighbor
interactions. The observed fractions are close to what would
TABLE VI. First shell reorientation times."
Ratio first bulk
Solute
LO
L+
LSO
S +!
S+
a
2.3
2.3
1.8
3.6
2.3
3.2
1.6
1.7
1.9
2.1
1.9
1.8
1.4
1.4
1.0
1.7
1.2
1.8
Integraltime constants [(T2H ), Eq. (3.2) 1ofthe I = 2 H-H vector reorientational correlation function [Eq. (3.1) 1 for the first solvation shell
( 1st) and bulk (B) water. All data are from simulations of clusters of size
256. Bulk values are averages over all molecules not within the first solvation shell or the surface region.
5053
be expected from a continuum calculation of the sort described here. For example, a region between the van der
Waals radius of L + and the first minimum in the L +
RDF contributes 71 % of the solvation energy of L + immersed in a homogeneous dielectric sphere of size appropriate to a 256 cluster. The observed fraction is 62%. It is interesting that such continuum estimates should yield answers
within - 20% of the observed values.
In Sec. V we will calculate the nonequilibrium response
. to a step change in solute charge assuming the solvation energy varies linearly with the size of the charge jump. By comparing Esw for the different solutes listed in Table IV we
might hope to determine what magnitude of charge jump is
consistent with this linear response assumption. From the
analysis of Sec. II it is possible to calculate the difference in
electrostatic energy between solutes that differ only in their
charge, based on solvent properties in the presence of either
one of the solutes. Denoting the two solutes 0 and b and their
charges qa and qb' the relation
Esw (0) - E.w (b)
=
(qa - qb) { (V) (b) -
:~ (I5V 2 )(b)}
(4.1 )
should hold if the linear approximation is valid. The quantities (V)(b) and (I5V 2 )(b) are the average electrical potential
and the square of its fluctuation in the presence of solute b.
(Values of these quantities are listed in Table VII.) Figure 6
compares the observed Esw differences to those calculated
on the basis of Eq. (4.1). There are 14 points on this plot
corresponding to 7 pairs of solute/cluster systems that differ
only in solute charge. [While there are only seven independent observed Il.Esw values there are twice this number of
calculated differences since either system can serve as the
reference in Eq. (4.1).] The agreement between the observed and predicted energy differences is fairly good. Such
agreement is not surprising when differences between large
solutes (ll.Esw between ± 100 kcaVmol) are considered,
since, as we have shown, the solvation structure about these
solutes is relatively insensitive to charge. The fact that the
observed Esw between S 0 and S + (extreme points) are also
within ± 20% of the linear predictions is quite surprising
however. The solvation structures about these two solvents
are very different. Furthermore, the per-molecule contribution of first shell waters to the solvation energy difference is
- 18 kcaVmol, which is larger than the water-water binding
energy in the bulk. In this sense the perturbation caused by
charging SO can hardly be considered small and the agreement with the linear prediction must be partly fortuitous.
A final aspect of the equilibrium solvation properties we
will consider is a more direct comparison of the observed
energetics to predictions of simple models of solvation. We
will later use these same models to compare to our dynamical results. The free energy of solvation of an ion is determined by the reaction potential as described by Eq. (2.6).
This reaction potential is simply the electrical potential at
the charge site that is due to the ion's polarization of its
surroundings. Observed values can be obtained from differ-
J. Chern. Phys., Vol. 89, No.8, 15 October 1988
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5054
M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
TABLE VII. Static electrical properties.•
First shell
Solute
N so1v
XIO
( V)
( Vzz )
xi()3
LO
LO
L+
L+
L-
496
236
496
240
236
507
252
123
59
251
252
-0.04
-0.02
- 1.06
-0.96
+0.98
-0.02
-0.01
-0.03
-0.02
- 1.24
-3.06
-0.02
2.16
2.08
- 2.54
-0.29
-0.77
-0.04
-0.44
9.92
34.3
SO
SO
SO
SO
S +!
S+
(lIV2)
X 10'
(liE;)
X 10"
0.92
0.85
1.04
0.97
1.10
2.21
2.01
1.88
2.08
3.44
4.03
2.68
2.48
3.03
2.82
3.20
16.9
16.1
17.3
17.5
47.5
62.7
(lIV;' )
(lIV2)
X 10"
X 10'
(liE;)
Xla"
0.354
0.482
0.526
0.622
10.6
10.7
11.8
12.3
46.4
90.0
0.71
0.96
0.94
1.03
1.83
1.69
1.84
2.05
3.56
3.93
4.5
4.9
5.7
5.6
6.3
17
27
22
23
61
76
a Average electrical potential ( V),
field component (Ez ), and field gradient component ( Vzz) and their fluctuations «lIV2) = (V2) - (V)2 etc.) in atomic units. The data correspond to results of particular simulations
as listed in Table IV.
b Contributions due to only first solvation shell waters.
ences between the (V) values listed in Table VII. We first
consider the solvent to be a homogeneous spherical dielectric with radius R and constant Eo. For such a model the
reaction potential of an ion of radius r0 and charge q is
1) (1 -R- .
q ( 1-VR(cont) = -- ro
ro)
(4.2)
Eo
Figure 7 compares the reaction potentials observed for the
ionic solutes to those calculated (X) from Eq. (4.2). For
these calculations a value of 218 was used for E061 and the
solute U radii for roo As illustrated by Fig. 7, the simple
continuum model does a good job of reproducing the observed potentials, coming within 13% for all of the ions studied. It is surprising that such a crude model should give this
level of agreement, especially for the small solutes for which
most of the potential comes from about ~ 7 molecules in the
first solvation shell. Also shown in Fig. 7 are the predictions
of the MSA model (0), which is based on a dipolar hardsphere representation of the solvent. 62 This model is of interest, not because it is especially accurate for equilibrium prop-
1
a
X-Continuum
O-MSA
(f)
-1
a:
>
-2
o
-3
-3
-2
-1
VR
VR (MSA)
= VR (cont)/(l + a),
(4.3)
where a is a correction term involving the solute/solvent
size ratio and Eo. Figure 7 shows that, compared to the simple continuum model, the MSA model yields better agreement to the observed potentials for the large solutes but considerably poorer agreement for the small solutes.
The success of the continuum model for describing the
reaction potential does not mean that the polarization surrounding the solute actually looks like that of a continuum
fluid. The polarization of a homogeneous spherical continuum surrounding an ion is
-q( 1 - -I)"
P eont (r) = - 41T
-::2'
Eo
(4.4)
r
In Fig. 8 we plot the ratio of the polarization observed to that
calculated on the basis of Eq. (4.4) for two solutes. The
actual polarization is highly structured and resembles the
water-solute radial distribution functions-quite unlike the
simpler
continuum dependence. The molecular, MSA
model accounts for just this sort of behavior,62 so that the
reason why the continuum model better reproduces the observed reaction potential is not obvious. We have recently
investigated continuum models for solvation dynamics
which are based on the idea that the dielectric constant
around a solute is not uniform but rather can be characterized by a radially dependent dielectric constant E(r).22 We
found that for reasonable choices of E(r) the dynamical predictions of such models closely resemble MSA predictions as
well as experimentally observed behavior. The data of Fig. 8
are rather noisy but tend to confirm one important result of
that work: the polarization and E(r) differ substantially
from homogeneous continuum predictions only in the region of the first one or two solvation shells. Thereafter, the
r
.0
0
erties63 but because of its recent extension 26-28 to the
dynamical problem to be discussed shortly. The difference
between the predictions of the MSA model and the continuum approach can be expressed as
a
1
Calc.
FIG. 7. Observed vs calculated reaction potentials (VR , a.u. X 10) of
charged solutes. Observed values are from Table V. Calculated values were
obtained for the simple continuum and MSA models using Eqs. (4.2) and
(4.3), respectively, as described in the text.
J. Chern. Phys., Vol. 89, No.8, 15 October 1988
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M. Maroncelli and G. R. Fleming: Dynamics
3
(A) L+
...., 2
c
0
o...U
'-..
1
0...
0
(8) S+
4
....,
c 3
0
o...U
'-..
0...
2
1
0
234
1
0
6
5
r/cr
FIG. 8. Solvent polarization surrounding ions (A) L + and (B) S +. Radial polarizations (1: i
r,) were calculated on the basis of point dipoles at
the water oxygen positions. The ordinate here is the ratio of the observed
polarization to that calculated for a dielectric continuum via Eq. (4.4).
",r
homogeneous continuum description is apt. However, as
was previously observed by Chan et al. 62 with respect to the
MSA model, no E(r) function everywhere greater than zero
is capable of reproducing the highly structured polarization
decays observed here. Thus, the present results warn against
viewing inhomogeneous continuum [E( r) ] models too literally.
v. DYNAMICS FROM EQUILIBRIUM SIMULATIONS
In Sec. IV we discussed the structural features of the
first solvation shells of the various solutes. The structure
imposed by the solute also alters the translational and rotational dynamics of first solvation shell waters from that of
bulk water. This effect is illustrated in Fig. 9 where we have
plotted the 1 = 2 correlation function of the H-H vector
o
+J
-1
u
c
-.J
-2
o
1
2
Time
3
(ps)
4
5
FIG. 9. Reorientational time correlation functions [C2H (1), Eq. (3.1) 1of
water molecules (512 cluster) as a function of radial shell relative to an SO
solute. The shells are defined by the radii (r/u): 1 = 0-1.60, 2 = 1.60-2.53,
3 = 2.53-3.47,4 = 3.47-4.40, and 5 = 4.40-6.00.
ot aqueous solvation
5055
[C2H (t), Eq. (3.1)] as a function of distance from an SO
solute (512 cluster). The average correlation times [Eq.
(3.2)] for shells 1-5 are 3.5 (first shell), 2.6, 2.4, 2.2, and 1.4
ps (surface), respectively. The reorientation of waters in the
first solvation shell of the SO solute are considerably slower
than in the bulk (1. 8 ps for pure clusters). The effect of the
solute on the dynamics of other shells is small, there being a
25% difference between the average value for interior shells
2-4 of 2.3 ps and the bulk value of 1.8 ps.
Table VI compares the average reorientation times
[ C2H (t), Eq. (3.1)] in the first solvation shell to the corresponding time in the "bulk" for all solutes studied (256 cluster results). Bulk in this table refers to all waters not in the
first solvation shell or in the surface layer. Results for the
small solutes are in accord with results obtained previously
by a number of groups.38-40 For both the hydrophobic and
hydrophilic extremes represented by SO and S + the reorientation times are roughly 1.5-2 times slower in the first
solvation shell than in the bulk. The S + ! solute exhibits a
faster, more bulk-like reorientation time for its first shell
waters compared to the S 0 and S + extremes. This intermediate case is close to the so-called "negative hydration" regime observed experimentally64 and in computer simulation 38 with ions of small charge/size ratio. In this regime the
solute exerts a structure-breaking effect on the first shell water and rotational and translational dynamics are speeded up
relative to bulk water. We have previously classified the
large solutes as being essentially hydrophobic in character.
The 40% slower reorientational correlation times exhibited
by L 0 and L + are in accord with this assignment. The L
solute appears to differ from the L 0 and L + in having a
faster first shell correlation time. The cause for such difference is not obvious from the differences in structure. This
observation is, however, in agreement with results for other
(smaller) anions in ST2 water simulations. 38
We now consider the time correlation functions that determine solvation dynamics within the linear response approach of Sec. II. These are TCFs of fluctuations in the electrical potential (c5V), field (c5Ez ), and field gradient (c5Vzz )
at the solute center and they correspond, respectively, to
solvation responses to step changes in the magnitude of a
centered point charge, dipole, or quadrupole as described by
Eq. (2.10). Figure 10 displays these three correlation functions observed with the SO and S + solutes. Figure 11 and
Tables VII and VIII summarize the results for all of the
solutes studied. The data and error bars shown in Fig. 10 are
representative of the results of these simulations. All three
correlation functions contain a very fast (- 25 fs) component that accounts for more than half of the total decay. This
component is a result of librational motions of water molecules. 65 We note that the presence of such a large amplitude,
fast component may be of prime importance in understanding reactions with high reactive frequencies in water. A longer component, which decays on alps time scale, is a result of
diffusive reorientations, and in some cases translational motions of solvent molecules (see below). For theSO molecule
[Fig. IO(A)] the decay of the potential TCF, (c5Vc5V(t), is
the most rapid with the relative ordering being given by
c5V>c5Vzz >c5Ez • With the exception of the S + solute
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M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
5056
1
TABLE VIII. Time constants of electrical TCFs.·
(Al SO
;!;. 0.5
u
1
;!;.
0.5
Solute
N so1v
(c5Yc5Y(t) )
(c5Ez c5Ez (t»
(c5Yu c5Yu (1»
La
La
L+
L+
LSO
SO
SO
SO
S +~
S+
496
236
496
240
236
507
252
123
59
251
252
0.07
0.11
0.14
0.16
0.21
0.12
0.09
0.10
0.09
0.20
0.23
0.14
0.19
0.17
0.25
0.21
0.18
0.16
0.17
0.10
0.33
0.15
0.13
0.14
0.22
0.19
0.15
0.14
0.14
0.08
0.30
0.13
u
o
o
0.1
0.2
0.3
Time
(ps)
0.4
0.5
FIG. 10. Time correlation functions of electrical properties observed for
solutes (A) SO (B) S + (256 clusters). The three curves in each panel
correspond to the normalized TCFs: solid = (c5Yc5Y(t»j dashed
= (c5Ez c5Ez (t»j and dotted = (c5Yu c5Yu (t». Error bars are estimates of
the statistical uncertainty due to finite simulation length.
shown in Fig. 10(B), forwhicht5V decayed the most sl~,,:ly,
the SO ordering is consistently observed for the remammg
solutes. In addition to its faster decay, in most cases the
(t5Vt5V(t) TCFs also show pronounced librational oscillations 67 largely absent in the (t5Ezt5Ez (t»
and
(t5V~t5Vzz (t» TCFs [Fig. 1O(A»). We will discuss reasons
for this behavior in Sec. VI.
Before discussing the dynamic differences observed for
the various solutes we first ask how the finite samples presented by these clusters might affect our results. It is useful
to calibrate expectations by again considering a continuum
so
LO
o
o
0.5
Time
0.5
(ps)
FIG. 11. Summary of the time correlation functions (c5Yc5Y(t» (solid
curves) and (c5Ez c5Ez (t» (dotted) for all solutes studied (256 clusters).
• Approximate time constants (ps) of the TCFs of the electrical potential,
field, and field gradient fluctuations [Eq. (2.10)]. These times are integrals of the normalized correlation functions between 0 and 1 ps. The correlation functions have not decayed completely to zero so that the values
listed are lower limits to the average time constants.
dielectric sphere as a model. The interaction energies of
point ion, dipole, and quadrupole solutes of radius r immersed in such a sphere having radius R are related to their
energies in an infinite medium by the fractions [1 - r/ R ) ,
[1 - (r/R)3), and [1 - (r/R)S). On this basis, Table V
lists values of the percentage of the bulk response expected
missing from cluster simulations using 64 to 512 molecule
clusters and the two solute sizes under study. Most of our
comparisons are based on 256 clusters. For such clusters,
Table V indicates that we miss -13% and - 29% of the
ionic solvation energy of Sand L size solutes, respectively.
For the ionic case, these values do not improve markedly
upon increasing the cluster size to 512. Solvation of the higher order multipoles are predicted to be much less influenced
by sample size. According to Table V, less than 3% of the
total dipolar solvation energy is neglected in 256 clusters,
and for quadrupole solvation the missing energy is at most
0.2%. Even for the worst case of an L solute in a 64 molecule
cluster the continuum approach predicts that at least 90% of
the full dipolar solvation energy should be accounted for.
From this exercise we conclude that our cluster results for
ion solvation dynamics may be somewhat influenced by the
sample sizes used. Deviations from the dynamics occurring
in bulk solution could be substantial for the large solutes but
should be quite small for the small solutes. Sample size probably has little observable effect on the calculated dipolar and
quadrupolar solvation dynamics.
Comparisons of the simulation results in Tables VII and
VIII for the same solute in different size clusters tends to
confirm the above expectations. In Table VII we list the amplitudes (t5V 2 ), (t5E;), and (t5V;) of the electrical properties. These t = 0 amplitudes determine the total change in
solvation energy in response to the three multi pole jumps.
Given that the statistical uncertainties in these quantities are
all in the range 5%-1 0% it is difficult to say with confidence
that differences as a function of cluster size, which are typically less than 10%, are significant. The approximate time
constants for decay of these TCFs, listed in Table VIII, show
much larger dependence on cluster size than do the amplitudes. Taking into account the roughly ± 30% uncertainties in these times, there still appears to be a significant differ-
J. Chem. Phys., Vol. 89, No.8, 15 October 1988
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M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
ence between the dynamics of L solutes in 256 and 512
clusters. For L 0 and L + all three correlation functions decay consistently faster in the larger size cluster. Based on the
continuum arguments given above, such behavior is not surprising for the (t5Vt5V(t) correlation function. However,
observing an apparently larger effect on the (t5Ez t5Ez (t»
and (t5Vzz t5Vzz (t» TCFs is unexpected. The direction in
which the rates change, getting faster with increased cluster
size, is consistent with current understanding of the solvation response (see below). In contrast to the L solutes, the
series of SO solute simulations shows little systematic variation of decay time with cluster size. The one exception is the
59 molecule cluster, which is actually too small to achieve
bulk-like behavior. (For this size cluster nearly all molecules
are part of the surface layer and the density, orientational
distributions, and reorientation times in even the first solvation shell are substantially different from their values in larger clusters. Thus, agreement with the larger cluster results is
not expected in this case and the simulation was run merely
to gauge how large the differences would be.) To summarize, we find that the dynamics calculated for the large solutes with clusters of size 256 and even 512 may differ significantly from the bulk dynamics. Without performing larger
simulations it is difficult to assess the magnitude of this effect, however, it seems unlikely that the time constants we
observe would differ from the infinite size limit by more than
a factor of2. The small solutes do not seem to be sensitive to
size effects for clusters in the range studied; thus, our observations for these solutes should directly reflect bulk dynamics.
Figure 11 illustrates the differences in solvation dynamics observed among the different solutes in clusters of size
256. For clarity we have only plotted the electrical potential
and field correlation functions. With the exception of S + ,
the field gradient correlation function lies between the two
TCFs that are plotted here. (The t5 Vzz TCF is also generally
similar in appearance to the t5Ez correlation function, as illustrated in Fig. 10.) Uncertainties in these C( t) plots are all
similar to those provided in Fig. 10. The time constants listed
in Table VIII serve to quantify the differences in the overall
decay times of these correlation functions. These times are
calculated as the integral under the C(t) curves as in Eq.
( 3.2) except that the integration is only carried out to t = 1
ps rather than to infinity in order to avoid the large errors
incurred by further integration. As can be seen from Fig. 11,
most of the C(t) curves have not decayed to zero by this time
and thus the values listed in Table VIII, although useful for
comparison purposes, are only lower limits to the actual time
constants of these correlation functions.
The main differences in the dynamics among the different solutes are as follows. The time constants of the uncharged solutes L 0 and SO are the same to within estimated
uncertainties, and the shapes of all three TCFs are also indistinguishable. (The amplitudes of the fluctuations, listed in
Table VII, are of course larger for the small solute.) Thus, a
factor of 2.25 in the solute/solvent size ratio has no observable effect on the solvation dynamics. We note that the similarity between SO and L 0 tends to argue against cluster size
appreciably affecting even the L solute results. The
5057
(t5Vt5V(t» correlation functions decay more slowly in the
charged solutes relative to the uncharged solutes. This is also
true of the (t5Ez t5Ez (t» and (t5Vzz t5Vzz (t» functions except
in the S + case. It is somewhat surprising that the solvation
dynamics for L + and L - should differ significantly from
that of L 0 since their solvation structures are so similar. In
fact, rather subtle differences in the structure about L 0,
L +, and L - appear to be responsible for the different
dynamics. For the large charged solutes we observe that a
small fraction (-4 out of 40) of the nearest-neighbor solvent molecules are persistently oriented so as to point an
appropriately charged interaction site at the solute. The dynamics of these particular waters is slower than the "normally" oriented waters and this seems to account for the difference in dynamics compared to the uncharged case. All
solvation TCFs of small charged solute S + ! decay much
more slowly than those of its uncharged counterpart. The
dependence of response time on charge is not simple however, and the more highly charged S + shows a similar response time for the potential TCF but times for the field and
field gradient TCFs which are as fast as in the uncharged
solute. As we have already described, the environment surrounding these small solutes changes appreciably as a function of charge so that such differences in dynamics are to be
expected. Some explanation for these differences will be discussed in Sec. VI. For now we note that the variations in the
collective TCFs do not correlate with the single particle
reorientation times given in Table VI.
It is useful to compare our results with the one published
study that provides the same sort of dynamical information
as derived here. 68 In their MD simulations of quadrupolar
NMR relaxation, Engstrom et al. 44 calculated the
(t5Vzz t5Vzz (t» correlation functions of Li+, Na+, and Clions in MCy69 water. From their data we estimate average
decay times of >0.16, > 0.25, and> 0.47 ps for these three
ions, respectively. None of our solutes have interaction parameters that correspond closely to any of these ions, however, based on solute-oxygen radial distribution functions,
we find that the size of N a + is just slightly smaller than our
S + solute. The value of > 0.25 ps for Na + may thus be
compared to our value of > 0.13 ps for S + .On this basis we
conclude that the two simulations are not in quantitative
agreement. The qualitative features of the C(t) decays reported by Engstrom et al. are, however, consistent with our
findings. For example, the correlation functions observed
for Li + , Na + , and Cl- all show a very rapid ( - 25 fs) decay
to less than half of the t = 0 value, and then a slower decay
with time constant somewhere in the few hundred fs range.
The trend of decreasing decay time with increasing charge to
size ratio is also in keeping with our findings for S + ~ and
S + . Lack of quantitative agreement between the two sets of
results can probably be attributed to differences in the water
models used. Engstrom et al. performed several of the above
simulations with varied system sizes of 64 and 125 molecules, and used both periodic (Ewald) and free cluster
boundary conditions. Since no significant difference in the
(t5Vzz t5Vzz (t» TCFs was observed due to these variations it
is unlikely that our two simulations would differ due to such
effects. On the other hand, characteristics of the two water
J. Chern. Phys., Vol. 89, No.8, 15 October 1988
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M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
5058
models are significantly different. The dipole reorientation
time of the MCY water model 70 is 2-3 times faster than the
ST2 value. Furthermore, the placement and magnitude of
negative charges in these models is quite different, which
would be expected to have a significant effect on the solvation of positively charged ions. Thus, the quantitative differences observed with these two water models are not surprising. Since we do observe this model dependence, caution
should be exercised in assuming that either sets of results
provide more than a semiquantitative guide to the behavior
of real water.
VI. COMPARISON TO THEORETICAL PREDICTIONS:
MOLECULAR ASPECTS
In this section we compare the solvation dynamics calculated from equilibrium simulations under the linear response assumption to predictions of two theoretical treatments. The first theory is the dynamic extensions- 12 of the
simple continuum model we have already used several times
for discussing the static energetics. This model treats the
solvent as a continuum dielectric fluid that is characterized
by its bulk dielectric dispersion, E( W ). The onset of a bulk
E(W) is assumed to occur abruptly at the boundary of the
solute, and thus the simple continuum model completely ignores any effects that the molecular nature of the solvent
might have on the dynamics. The second and more recent
approach, the dynamical MSA model,2I,26-28 does consider
molecular effects through use of a hard-sphere solute and a
dipolar hard-sphere solvent as a reference system. The static
behavior of such a model system can be solved within the
mean spherical approximation (MSA) and can then be used
to predict solvation dynamics for a real solvent. Loosely
speaking, the dynamics are obtained by replacing Eo in the
static reference model by E(W) of the solvent of interest. Predictions of the dynamical MSA model depend on E( w) and,
in addition, on the single molecular parameter p, the ratio of
the solvent and solute sizes.
In order to compare the predictions of these two models
to our simulation results, the dielectric dispersion of the ST2
water model is needed. In the Appendix we describe an approximate cluster calculation of E( w). Here we will use preliminary results of an extensive investigation by Neumann 61
since these should be more accurate. Neumann's results
(293 K) can be expressed in terms of a triple Debye fit as
E(W)
= Eoo3 +
3 EOj - E ooj
L
---=--~
j=
1 + iWT
1
(6.1 )
Dj
using the parameters
= EOI = 218, TDI = 33.16 ps,
E oo1 = E02 = 2.73, TD2 = 0.177 ps,
E oo2 = E03 = 1.79, T D3 ~0.013 ps,
cept at very early times. The dielectric constant ofST2 water
is much too high compared to the experimental value of78.4
(298 K, Ref. 57), in keeping with the general observation
that ST2 water tends to overestimate the extent of ordering
present. 30 The Debye time of the main dispersion T D 1 is also
quite long compared to the experimental value of 8.3 pS.S7
However, the quantities of most relevance for solvation dynamics are the longitudinal relaxation times. For the main
( 1) dispersion regime, the longitudinal relaxation time
[TLI = (Eool/Eol)TDI] of this model is 0.42 ps (or 0.49
pS71) compared to an observed value of 0.54 ps. Thus, although the ST2 potential yields rather poor estimates of EOI
and T D I' predictions of solvation dynamics based on this
model are expected to be much more accurate.
Figure 12 illustrates the solvation response functions
Set) calculated from the simple continuum, and dynamical
MSA models using the above E(W). In Fig. 12(A) the response to a change in solute charge, i.e., the (8V8V(t» response, is shown. The simple continuum model predicts a
response that is independent of solute size or charge. The
MSA prediction does depend on the solvent/solute size ratio
and the two curves shown correspond to values approximate
to Sand L solutes as marked. All three S( t) curves exhibit a
fast initial decay that arises from the librational (TD 3 ) component of E(W). The longer time part of Set) is due to the
true relaxation terms (TD1 , T D2 ). Within the simple continuum model these two dispersions produce a biexponential
Set) which after -0.5 ps becomes nearly monoexponential
with a time constant of 0.49 ps. In contrast, the S(t) curves
predicted by the MSA model are nonexponential over the
time window shown, and decay much more slowly than the
continuum model. These features are characteristic of the
0
(A)
-1
-2
5
- -3
c
L
~
rJJ
...J
-4
Cant.
a
-1
+.J
-2
rJJ
- -3
c
...J
-4
Eo
Eoo = Eoo3 =
1.
In this expression we have approximated the librational resonanceobserved by Neumann (w-2X 10 14 s- l ) in terms of
a third relaxation regime in order to simplify the theoretical
calculations. The main effect of such an approximation is to
smooth out any oscillatory behavior in the calculated response without significantly changing its overall shape ex-
-5
a
0.5
Time
1
1.5
2
(ps)
FIG. 12. Solvation responses functions predicted by the continuum and
MSA models (see the text). (A) Response to a step function change in
solute charge. The MSA prediction depends on solute size and the curves
labeled Sand L correspond to the two solute sizes simulated. (B) Responses
to step function changes in solute charge (n, dipole moment (Ez )' and
quadrupole moment (Vzz) predicted from the continuum model. These
S(t) curves are labeled according to the equilibrium TCFs to which they
correspond according to Eq. (2.10).
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M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
predictions of the MSA model even for single Debye E(Cd).
The average time constants [Eq. (3.2)] ofthe three curves
in Fig. 12(A) are: simple continuum (r) = 0.15(0.13) ps,
MSAL (r) = 0.94(0.24) ps, and MSAS (r) = 1.7(0.33)
ps. The numbers in parentheses are the values obtained when
the integral over S(t) is truncated at 1 ps, and are given for
comparison with the simulation results listed in Table VIII.
Figure 12(B) shows the continuum model predictions
for the response to jumps in a solute's charge, dipole moment, and quadrupole moment. The continuum model predicts that the solvation times increase in the order ion < dipole < quadrupole with average response times [Eq. (3.2)]
of 0.15, 0.23, and 0.25 ps, respectively. The long time behavior of all three of these responses is similar; the limiting
slopes in Fig. 12(B) give time constants of 0.49, 0.55, and
0.57 ps. The same relative ordering of response times should
also hold for the dynamical MSA model, however, the differences between the multipoles are predicted to be larger. Unfortunately, the dynamical MSA model has not been solved
for higher multipoles with arbitrary E( Cd) except in the case
of a dipolar solute of size equal to that of the solvent. 27(b).72
For the one available comparison, that of a dipolar jump in
anSsolute, the MSAmodeI24 (b) predicts (r) = 5.1(0.63) ps
compared to the 1.7(0.33) ps for a charge jump.
In Fig. 13 we plot the simulated C(t) TCFs in the same
manner as the theoretical predictions. In order to obtain the
best signal to noise in these logarithmic plots we have averaged together results of several simulations. Since the uncharged solutes L 0 and S 0 exhibit C (t)'s that are identical to
within our uncertainties, we combined results of these solutes in producing Fig. 13. (c5Vc5V(t» correlation functions,
to be compared with the ionic S(t)'s in Fig. 12(A), are
shown for the solutes S 0/L 0, L +, and S + in Fig. 13 (A).
One point, which was obvious even before carrying out the
0
-1
...,
-2
2. -3
c
.J
-4
0
-1
...,
-2
2. -3
c
.J
-4
-5
0
0.5
Time
1.5
1
2
(ps)
FIG. 13. Electrical time correlation functions observed for several solutes
plotted for comparison to the theoretical predictions in Fig. 12. (A)
(8 V8 V( t) ) TCFs for solutes S + ,L +, andSO/La. These curves were obtained by averaging results of several simulations. For the SO/La curve results for SO and LOsolutes were combined. (B) (8Yc5V(t», (8E.8E.(t»,
and (8Vzz 8V.. (1» TCFs for SO/La solutes.
5059
theoretical calculations, is that neither model is able to account for the variations we see in the (c5Vc5V(t» TCFs of the
different solutes. The simple continuum model predicts that
(c5Vc5V(t» (and the other TCFs) should be the same for all
solutes. In fact, we observe that they decay with time constants that differ several fold. The dynamical MSA model
does allow for some differences in the behavior of different
solutes but the only solute attribute accounted for is size. In
contradiction to the MSA prediction [Fig. 12(A)], we find
that over the range used here, size is of little importance. It is
rather the charge/size ratio that appears to determine the
dynamical differences seen among these solutes. With this
underlying difficulty in mind, we note that (c5 Vc5 V(t) ) of the
uncharged solutes is close to the prediction of the simple
continuum model, except for being slightly faster at early
times. The dynamical MSA model predicts decays that are
too slow compared to any of the observed (c5Vc5V(t» TCFs.
Figure 13(B) illustrates the observed ordering of the potential, field, and field gradient TCFs using the uncharged solutes as an example. Although the potential TCF does decay
the most rapidly, as predicted by both theories, the ordering
of the other two TCFs is reversed from that predicted. The
long time behavior of the (c5Ez c5Ez (t» and (c5Vzz c5Vzz (t»
TCFs is slower than that of (c5Vc5V(t» or the predictions of
the continuum model and approaches the sort of decays predicted by the MSA theory. However, no TCFs were observed to decay as slowly as the MSA dipole prediction (not
shown).
To summarize, neither the simple continuum model nor
the dynamical MSA model is able to quantitatively reproduce the observed behavior. The best that can be said is that
the predictions of overall time scales appear to be within
about a factor of 2 or 3 of the observed values. The reasons
for lack of predictive ability of these theories arises, at least in
part, from the highly structured nature of aqueous solvation.
We now investigate some of the dynamical consequences of
this structure.
In Fig. 14 we consider how the total solvation response
is built up from contributions of different solvent regions. In
this figure we concentrate on the response to a change in
solute charge as determined by the (c5Vc5V(t» correlation
function. For an example we will use the results of a simulation of the SO solute in a 256 cluster, however, our observations are general. In Fig. 14(A) we plot contricutions to
(c5Vc5V(t» from a series of shell regions 1-3 whose definitions are provided in Table IX. Also shown in Fig. 14(A) is
the single particle reorientational TCF of the dipole moment
direction [C1d (t), Eq. (3.1)] for molecules in the first solvation shell. The contribution made by anyone molecule to
(c5Vc5V(t» decays on a time scale similar to this reorientational TCF. 73 The difference between the decay times of
(c5 Vc5 V( t) ) and this single particle time is striking and serves
to highlight the fact that the solvation response is as fast as it
is because it results from the concerted motion of many molecules. The number of molecules actually required to
achieve the full cooperativity of the response appears to be of
order 10-40, based on the fact that both the first and second
shells exhibit this fast decay. A second important feature
illustrated by Fig. 14(A) is how important the first shell
J. Chern. Phys., Vol. 89, No.8, 15 October 1988
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5060
M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
(8Y8 V(t» TCF, the shell contributions to (8V8V(t) )
should be related to the shell polarizations as
(A)
(8Y8V(t»
(8V2)
::'0.5
U
(8Pr 8Pr (t»
(8P;)
(6.2)
where Pr is the radial component ofthe polarization
(6.3)
(8)
::. 0.5
U
'\
'~'''.
~':':::">,::::.::::,:...................~'I-~~..--..j
o
o
0.5
Time
1
(ps)
FIG. 14. Electrical time correlation functions for an SO solute decomposed
into individual shell contributions. (A) (BVBV(t». The solid curve at the
bottom is the total TCF and the curves labeled 1-3 are the contributions of
individual shells 1-3. (Definitions of the shell regions are provided in Table
IX.) The curve marked p. is the single-particle dipole TCF [C1d(t),
Eq. (3.1) 1shown for comparison. (B) Correlation functions of the radial
polarizations [ Pr , Eq. (6.2) 1 of the shell regions.
contribution is to the total response. The I = 0 amplitude
( (8y2) ) contributed by the first solvation shell is 85% of the
total (see Table VII). Furthermore, the large oscillations as
well as the rapid decay of the potential TCF are due predominantly to the behavior of first shell molecules. For the SO
solute the first shell contains -16 molecules. If one examines the potential correlation function for only half this number of first shell molecules, the oscillations as well as the
rapidity of the response decrease markedly. Thus, both effects appear to be a collective libration of all molecules in the
first solvation shell.
A further aspect of the first shell contribution is that it is
sensitive to the finite size of the solvent molecules in a way
not reflected in E(m). The dielectric response ofa liquid depends only on reorientational dynamics of the molecular dipole moment. To the extent that this is also true of the
TABLE IX. Shell definitions used in SO and S
Shell No.
'in b
1
2
3
4
0
1.60
2.45
3.30
rout
+
b
I
2
3
4
0
1.10
2.20
3.30
1.60
2.45
3.30
5
+
solute
1.10
2.20
3.30
5
• 256 molecule clusters.
Inner and outer shell boundaries in units of the ST2 u.
C Average numbers of molecules in shell.
b
(A)
3
2
::'0.5
U
(8)
simulations.'
No. molsc
::'0.5
SO solute
S
of a shell. [The summation in Eq. (6.3) extends only over
molecules in a given radial shell.] Figure 14 (B) shows the
polarization TCFs as a function of shell for the same SO
simulation as in Fig. 14(A). In contrast to (8V8V(t», the
(8Pr 8Pr (t» TCF of first shell waters actually decays more
slowly than the other shells. Note also that the first shell
polarization shows little of the oscillatory behavior of the
potential TCF. The second and third shell polarizations decay with essentially the same time dependence as their contributions to (8V8V(t». This latter observation is to be expected since, except for the innermost shell, interactions
between the solute and solvent should be well represented
treating the solvent as point dipoles. Lack of agreement
between (8Pr 8Pr (t» and (8V8Y(/» in the first solvation
shell indicates that here the solute does not view solvent molecules as simple point dipoles but rather interacts with their
extended charge distribution, in this case the four point
charges of the ST2 model. A second cause for the difference
could also be that translational motions of molecules in the
first shell significantly contribute to the response. That is
since Va: PJr, changes in r as well as changes in Pr can affect
the (8V8V(/» decay. As will be discussed presently this
latter effect appears to be small on the potential TCFs of
uncharged solutes.
Figure 15 illustrates the shell contributions to the elec-
16.4
43.4
83.1
109.1
U
1+2
1+2+3: Tot
o
o
6.9
39.4
101.8
103.8
0.5
Time
1
(ps)
FIG. 15. Shell contributions to the (BE.BE.(t» TCF of an SO solute: (A)
individual shell contributions and (B) sums over individual shell contributions. Note that in panel (B) there are two curves at the bottom of the plot
that are nearly superimposed-that of the (6E.BE. (t» contribution from
regions 1-3 and that ofthe total (1-4) TCF. Definitions of the shell regions
are provided in Table IX.
J. Chern. Phys., Vol. 89, No.8, 15 October 1988
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M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
tric field TCF, (8Ez 8Ez (t», for the same SO simulation.
The contrast between the behavior of this field correlation
function and the potential or polarization TCFs just discussed is remarkable. As seen from Fig. 15(A), after an initial fast librational decay, the contributions from any given
shell decay very slowly. It is only when contributions of
successive shells are summed [Fig. 15 (B) ] that the response
achieves the very fast rate characteristic of solvation. Such a
difference between the behavior of the potential and field
TCFs is surprising. At first sight, it appears that to obtain the
fast solvation response many more molecules are required in
the (8Ez 8Ez (t» TCF than in the (8V8V(t» TCF. Since
the range of the field is much smaller than that of the potential, this result is counter intuitive. The problem is in some
sense an artifact and involves the difference between the scalar and vector character of Vand E. We note that the way in
which the shell contributions to (8Ez 8Ez (t» decay is sensitive to the exact placement of the shell boundaries, a feature
not observed for (8V8V(t». What causes this sensitivity is
the fact that solvent molecules are not spherically distributed about the solute. For a shell of molecular width, at any
given time one is likely to find, e.g., an "extra" solvent molecule present in the + z direction as compared to the - z
direction. The electric field produced by molecules in this
shell, especially the E z component, will notice this directional asymmetry. The potential, on the other hand, will only be
sensitive to the net number of molecules in the shell and not
to any directional biasing. It seems reasonable to assume that
the spherically averaged density of a shell should fluctuate
less, and on a faster time scale than fluctuations at any particular point within the shell which should proceed on a diffusional time scale. Thus, shell contributions to
(8Ez 8Ez (t» include components due to relatively slow diffusion in and out of the shell regions much more so than does
the spherically averaged (8 V8V(t) ) TCF. Only when these
diffusional components are suppressed by summing shells
together (i.e., making them larger) are decays similar to the
(8V8V(t» shell contributions obtained. The above arguments would imply that the shell contributions to
(8Vzz 8Vzz (t» should probably be more similar to the
(8Ez 8Ez (t» TCFs than to those of (8V8V(t». Unfortunately, we do not have such data available for the field gradient components.
The last aspect of the equilibrium TCFs that we will
consider is the molecular basis for the differences observed
among the charged and uncharged solutes. We first note, as
expected, these differences are due almost entirely to differences in first solvation shell dynamics. For example, the potential and polarization TCFs ofSO and S + are the same to
within uncertainties for all but the first solvation shell. Figures 16 and 17 contrast the behavior of molecules in the first
solvation shells of S 0 (Fig. 16) and S + (Fig. 17) solutes.
What are plotted here are three observables of the trajectories of a single solvent molecule. In both figures panel (A)
shows the solute-oxygen distance, panel (B) the cosine of
the angle between the solvent dipole and the radius vector,
and panel (C) the single-molecule contribution to the electrical potential at the solute center. Comparison of panels
(A) and (B) on either figure shows that small scale rota-
(A)
1.4
.,
5061
:l
.....
'C
tl
1. 2
o
1 ~~--~-------------T--~
0.6
"-
*
!3
0.1
en
o
u -0.4
-0.9 ~~--------------------~
."
2
....
:ii
....o
0:
0
-2
w
o
1
0.5
Time
(ps)
FIG. 16. Properties of the trajectory of a single solvent molecule in the first
shell ofan SO solute: (A) solute-oxygen distance; (B) - cos( fL·") where
fL is the dipole direction and" the solute-oxygen vector; and (C) the singlemolecule contribution to the electrical potential.
tional motions occur with roughly four times the frequency
of translational motions. 74 These two motions contribute
with different weights to the potential fluctuations in the two
solutes and are responsible for the differences between their
(8V8V(t» decays. For the SO solute, Fig. 16 shows that
fluctuations in the potential [Fig. 16 (C)] arise mainly from
rotational motions, overall molecular translation has little
affect on the potential. This is not the case for the S + solute
for which translational motions do play an important part in
the potential fluctuations. Waters in the first solvation shell
of S + point one oftheir negative charges (Q1) directly at
(A)
.,
:l
."
'C
IC
II:
00.8
-0.4
.t,
S -0.6
en
0
u -0.8
-1
....
IC
."
....c
....0
0:
-4
-6
Q)
-8
w
-10
0.5
0
Time
1
(ps)
FIG. 17. Properties ofthe trajectory of a single solvent molecule in the first
shell of an S + solute (see caption to Fig. 16).
J. Chem. Phys., Vol. 89, No. 8,15 October 1988
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5062
M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
the solute and are held rather tightly. The tight binding is
manifest in the coherent vibrational character of the soluteoxygen distance shown in Fig. 17(A). [This coherence is
also responsible for the unique oscillation at - 0.18 ps in the
S + TCFs shown in Fig. 1O( B).] In the "pointing" configuration, QJ is close to the solute center, and it is mainly the
solute-QJ distance that determines the potential. Fluctuations in this distance result primarily from translational
(/vibrational) motions rather than rotational motions as is
illustrated in Fig. 17. That is, while the rapid spikes in Fig.
17 ( C) do come from motions that are rotational in character, the overall contour of the potential trajectory closely
matches excursions in the oxygen (essentially the center of
mass) to solute distance. Thus the much slower (8V8V(t»
response of the S + solute compared to the SO solute is a
result of the importance of slower, translational mechanisms
of relaxation in the former and not the latter solute.
The observations made on the basis of Figs. 16 and 17
can be generalized to the other solutes and responses. As
expected, the large uncharged solute behaves in much the
same way as does SO. The S + ~ solute seems to have important contributions from both rotational and translational
mechanisms. As already discussed, the first shell structure
surrounding this solute is a mix of the structures observed
for SO and S +. The fraction of nearest neighbors in the
pointing configuration contribute to the decay of (8 V8 V( t) )
mainly via their translational dynamics whereas the other
first shell solvents appear to have mainly rotational contributions. The case for the L + and L - solutes is similar
although less clear cut. We do observe that a small fraction
of the nearest neighbor molecules of these solutes adopt
pointing configurations. These molecules appear to be responsible for the slower overall response, however, it is difficult to say that they contribute mainly via a translational
mechanism, since with the large solutes the effect is considerably blurred, and reorientational motions are equally important.
We have also investigated how the single-molecule dynamics affect the higher order, (8Ez 8Ez (t»
and
(8 Vzz8 Vzz (t) ) TCFs. Comparisons like those shown in Figs.
16 and 17 for the potential are similar in these latter cases.
One difference is that there is a less direct relation between
fluctuations in the field and field gradient and the dipole
angle for S 0 than is shown in Fig. 16. Since solvent molecules
are not point dipoles, the dipole direction only partially
specifies the electrostatic interaction with the solute for
small separations. (This effect is responsible for the difference between the polarization and potential TCFs described
earlier.) The difference should be least important for the
potential and become progressively more noticeable for the
progressively shorter range field and field gradient properties. Thus the above observation is to be expected. For the
same reasons we would anticipate translational effects to become of increasing importance in this same order. This expectation does seem to be born for theS + solute where, e.g.,
we observe that the low-frequency oscillations in the solvation time correlation functions caused by translational motions increase in amplitude in the order (8V8V(t»
< (8Ez 8Ez (t» < (8Vzz 8Vzz (t» [Fig. IO(B)]. The high-
frequency oscillations in these TCFs, due to librational dynamics, decrease for all solutes in this same order. Thus, it
appears that differences in the three multipole response
functions, which parallel the differences observed in
(8V8V(t» in the progression SO .... S + ! .... S +, may also
reflect the increasing relative importance of translational relaxation mechanisms.
VII. NONEQUILIBRIUM SIMULATIONS
In Secs. V and VI, we have examined the time dependence of the solvation response using results of equilibrium
MD simulations. The TCFs of electrical properties observed
in equilibrium are simply related to the nonequilibrium solvation response [via Eqs. (2.10)] only to the extent that the
linear response approximation is valid. Thus, if an SO solute
in equilibrium is suddenly given an increment of charge 8q,
the subsequent relaxation of the solvation energy follows the
same time dependence as (8V8V(t» evaluated with SO in
equilibrium, as long as 8q is sufficiently small. The question
remains as to just how small the charge (or dipole, or quadrupole) change must be for the linear response approximation to hold good. In Sec. IV we observed that the static
energies of solvation of ions behaved in a linear manner over
an unexpectedly large range of charge. In order to explore
what happens for the dynamics, we have performed a number of nonequilibrium simulations in which we directly monitor the solvation response to a change in solute charge. The
charge jumps we simulated are listed in Table I and correspond to changes among the various solutes that were studied in equilibrium. From such simulations we derive the normalized function S(t) of the electrical potential [V(t)]
response to the jump (at t = 0):
S(t)
=
V(t) - V( (0) .
V(O) - V( (0)
(7.1)
Comparison of S(t) to the (8V8V(t» TCF observed in
equilibrium allows us to assess the range of validity of the
linear approach for ionic dynamics.
One aspect of these nonequilibrium simulations must be
mentioned before making comparisons to the equilibrium
results. When the charge on a solute is abrutply changed the
difference in solvation energy appears as added thermal energy that produces local and, for finite samples, global temperature changes. This effect was first observed by Rao and
Berne45 in a simulation of charge jumps, Ne .... Ne+ 2 , in ST2
water (the Ne U parameters in this simulation were identical to our SO parameters). For this drastic change in the
solute's charge, Rao and Berne observed a dramatic local
heating of first solvation shell waters to rotational temperatures of -1200 K. The tightly bound first coordination
sphere was observed to retain much of this high thermal
excitation for the 0.6 ps duration of their simulation. 45 As
illustrated in Fig. 18, we see a similar, although less dramatic, heating in our charge-jump simulations. In this figure we
plot temperatures, determined from the average kinetic energy per molecule in a given shell, as a function oftime after
the charge jump. (Shell definitions are provided in Table
IX.) The largest perturbation we have examined is the
SO .... S + change shown in Fig. 18(A). In this case the kinet-
J. Chern. Phys., Vol. 89, No.8, 15 October 1988
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5063
M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
700
(A) 50-75+
g 600
OJ
t.
B 500
't."
OJ
~400
OJ
I-
300
(8) 50-75+~
_ 450
~
OJ
5 400
+J
'"
t.
OJ
~ 350
OJ
I-
300
0
0.5
Time
1
(ps)
FIG. 18. Temperatures of different solvent regions as a function of time
afterachargejump (att = 0): (A) SO .... S + and (B) SO .... S + !. Temperatures are based on the total kinetic energy of molecules in a region. The
dashed lines indicate the temperature prior to the charge jump. Definitions
ofthe shell regions are provided in Table IX.
ic energy of the - 7 first solvation shell waters increases by
an amount equivalent to a 300 K temperature increase. The
heating is very rapid, the majority occurring in one librational period. All solvent regions show this abrupt temperature increase, which is presumably communicated from the
first shell through the librationallvibrational modes of the
hydrogen bonded water network. Relaxation of the majority
of the excess first shell energy to the remaining solvent, however, takes place on a much longer time scale of order 1 ps.
As a result of the SO -+ S + charge jump, the average total
kinetic energy of the 256 cluster rises an amount equal to a 39
K temperature change. With the exception of the reverse
(S + -+SO) jump, local heating was much smaller in all of
the other nonequilibrium simulations undertaken. The next
largest effects were seen with the S 0 -+ S + ~ case, illustrated
in Fig. 18 (B). With this smaller charge jump, the kinetic
energy rise in the first shell is reduced to the equivalent of
-100 K. By the end of the simulation about 75% of this
energy has flowed into the remaining solvent. The overall
temperature rise in this case is only - 3-5 K. Jumps of a full
electronic charge in the L solutes are similar to the
SO-+S + ! case in that the overall temperature rise in the
sample is again - 3.5 K. The main difference is that, since
the number of first solvation shell waters of the L solutes is
roughly six times that surrounding S + ~, the initial kinetic
energy of temperature change per molecule in the first shell is
reduced accordingly to - 20 K.
How will this local heating influence the dynamics and
comparison to the equilibrium results? First we note that
such heating is one feature present in the nonequilibrium
situation that might cause departure from linear behavior.
We note that finite sample size should not appreciably bias
our observation of this effect since the heat dissipation is
relatively slow on the solvation time scale. It would be rea-
sonable to assume that local heating would hasten the solvation response. Some idea of the expected magnitude of the
change can be obtained by noting that reorientation times
determined from NMR measurements7S decrease by factors
of 1.8 and 4.3 as a result oftemperature increases of 20 and
80 K, respectively. On this basis, even the fairly modest heating (20 K) of first shell waters produced by a full charge
jump in an L solute might be expected to increase the solvation rates noticeably (i.e., twofold) compared to the equilibrium predictions. Via this same reasoning, the - 300 K
change in energy of first shell waters in the S 0 -+ S + simulation would be expected to produce an enormous change in
the observed rates. Here, however, the idea of a local temperature is somewhat misleading. Most of the excess energy
is tied up in vibrational energy of the strongly bound first
shell-solute complex that forms very rapidly upon ionization. We observe that the amplitudes of the translational/
vibrational modes such as illustrated in Fig. 17 (A) increase
while their frequencies remain about the same. This vibrational sort of heating would not be expected to enhance the
rate to nearly the extent that an actual temperature change,
which serves to speeding up all diffusive motions, does.
Thus, while we would expect local heating to have some
effect on the SO -+ S + response it should not be as extreme
as the 300 K figure might suggest. Finally, we note that the
ST2 model does not allow for intramolecular vibration of the
solvent. The additional heat capacity introduced by such
modes should further dampen the effect of local heating in
real water compared to what we observe in the simulations.
The results of the charge jump simulations are summarized in Table X and in Figs. 19 and 20. In Fig. 19 we compare
the normalized potential response functions S(t) [Eq.
(7.1)] for two charge jumps (A) L O.... L + and (B) SO
-+S + to the (I5VI5V(t) TCFs [C(t)] observed for these
solutes at equilibrium. Consider first the L O-+L + case
shown in Fig. 19(A). At early times, the S(t) response is
identical, to within uncertainties, to the C(t) correlation
function of the L 0 solute. At longer times (t > 0.3 ps) S ( t)
reaches its ultimate value significantly faster than either of
the equilibrium CU) decays. All four jump simulations inTABLE X. Time constants of the nonequilibrium solvation response.
Equilibrium timesb
Jump
Nonequilibriuma
response time
LO .... L +
LO .... LL +! .... L +
L + .... LO
SO .... S!
SO .... S +
S
+ .... SO
0.07
0.11
0.12
0.08
0.16
0.21
0.04
Initial
Final
0.11
0.11
0.16
0.21
0.16
0.11
0.20
0.23
0.09
0.16
0.09
0.09
0.23
a Approximate average
response times (ps) ofthe electrical potential to the
indicated charge jumps. These times are integrals of the normalized response functions [Eq. (7.1) I between 0 and 1 ps. Not all responses have
decayed to 0 by 1 ps so that the values listed are lower limits to the average
time constants.
b Approximate average time constants (ps) of the (.5V.5V(t»
TCFs observed in equilibrium for initial and final states of the jump. Data taken
from Table VIII.
J. Chern. Phys., Vol. 89, No.8, 15 October 1988
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M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
5064
u 0.5
(fJ
o
(8) SO~S+
u
.0.5
I~.
t,\~ 0,. \
oJ\,.,..,.
V
(fJ
v
1\." '-,.
y
o
o
\ . \ ' \............",......................
0.5
Time
1
(ps)
FIG. 19. Response functions of the nonequilibrium potential response to a
in Eq. (7.1)] compared to (8VcW(t»
charge jump [solid curves;
TCFs observed in equilibrium at the initial and final states of the jump: (A)
LO .... L + jump and (B) SO .... S + jump.
set)
volving L solutes exhibit this same basic pattern. At short
times the S( t) responses are identical to one another and to
C( t) of L 0 at equilibrium. The behavior at longer times can
be judged from the approximate time constants listed in Table X. Although the uncertainties in these values are large,
(roughly ± 30%), the S(t) curves for L solutes consistently decay at least as fast, and usually faster than the equilibrium TCFs.
Differences among the S solutes in equilibrium are
much larger than among the L solutes so that comparisons
to the nonequilibrium simulations are more clear cut. For
theSO-S + case illustrated in Fig. 19(B),thenonequilibriumS(t) curvemorecloselyresemblesC(t) oftheS + solute
than that of SO. The reproducible maximum at -0.2 ps in
C(t) of S + is not found in theSO-S + S(t) decay, however, the decay times are quite similar (Table X). Further, as
in the equilibrium C(t) ofS +, theSO .... S + S(t) lacks the
pronounced librational oscillations present in the SO TCF.
1
;;0.5
en
o
o
0.5
Time
1
(ps)
FIG. 20. Comparison of the potential response to the opposite chargejumps
S + .... So (solid) andSO ..... S + (dashed).
In an analogous way the SO ..... S +! response function (not
shown) is more similar to the S + ! equilibrium TCF than to
that of SO. The average time constants for the SO ..... S +!
[S(t)] and S +! [C(t)] decays are the same to within our
estimated uncertainties.
The response observed with the S + ..... S 0 charge jump
was unique among the simulations we have performed. Its
S(t) curve is shown in comparison to that of the reverse
jump (SO ..... S + ) in Fig. 20. Whereas all other S(t) responses resemble one or other of the equilibrium TCFs, the
S + ..... SO response is much faster than any equilibrium decay. Most of the relaxation is over within one librational
period. The mechanism in this case involves first solvation
shell waters undergoing an impulsive reorientation and
translation away from the solute. The reason is that the coordination sphere of the initial S + solute is so tightly bound
that there is considerable repulsion between neighboring waters, which have further unfavorable LJ interactions with
the solute. When the charge is switched off this repulsion is
bared and causes very rapid expansion of these waters. None
of the other situations investigated has such a switching on of
repulsive forces to nearly the same degree as the S + ..... So
case.
A number of conclusions can be drawn from the above
results. First of all, jump of a full electronic charge in either
the L or the S solutes cannot strictly be described in terms of
a linear response. This fact was already apparent from the
eqUilibrium simulations in which we observed that the equilibriumTCFsofSOandS + ,and even ofL O,L + ,andLwere perceptably different. However, in a practical sense we
could categorize the behavior of the L solutes with respect to
a full charge jump as "nearly linear." Differences among the
TCFs of the different L solutes in eqUilibrium are not large.
The charged solutes do decay more slowly at long times but
the early time behavior [where most of the C(t) decayoccurs] is quite similar in all cases. Furthermore, in the nonequilibrium simulations we actually observe no difference
between the responses to the opposite charge jumps L 0
..... L + and L + ..... L 0, and the differences among the other
L solutes do not exceed our uncertainties. Finally, all nonequilibrium response functions closely resemble the equilibrium TCFs, especially that of L O. The main difference is
again in the long time behavior where the nonequilibrium
response appears to be more rapid. We speculate that this
slightly faster response is an effect of the local heating discussed earlier. The behavior of the S solutes is more complicated. We would not expect the solvent response to a full
electronic charge jump in the S solutes to be well described
by a linear response approach. The solvation structures of
theSO and S + solutes are very dissimilar, as are their equilibrium dynamics. The dramatic difference between jumps
in the SO ..... S + and S + ..... SO directions upholds this
expectation. Even here, however, the linear response formalism still seems to have some predictive value. Examination
ofall of the nonequilibrium results (Table X) shows that the
linear response prediction based on the final state of the solute yields an estimate of the response time correct to within
about a factor of 2. This estimate appears to form an upper
bound to the observed response time, possibly because of
J. Chern. Phys., Vol. 89, No.8, 15 October 1988
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M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
local heating effects. With the exception of the somewhat
pathological S + . . . SO jump, the shapes of the S(t) curves
are also reasonably well predicted by the linear response
C(t) ofthe final state.
VIII. SUMMARY AND CONCLUSIONS
We have used molecular dynamics computer simulations to study the dynamical aspects of solvation of simple
spherical solutes in ST2 water. Although we employ solvent
clusters in these simulations, we have shown that for the size
clusters employed, we are observing behavior characteristic
of the bulk solvent. Some sample size effects are noticeable in
the dynamical results, however, our qualitative conclusions
should be independent of such effects. We now summarize
our main results.
The equilibrium simulations yield the time correlation
functions (6V6V(t», (6Ez 6Ez (t», and (6 Vzz 6 Vzz (t»,
whose time dependences are those of the solvation responses
to small step function changes in the charge, dipole, and
quadrupole moment of a solute. From these simulations we
observe the following:
(i) All of the TCFs show a very rapid (-25 fs) librational decay accounting for roughly half of the total relaxation. The remainder occurs more slowly, on the time scale of
tenths of picoseconds.
(ii) The relative rates of decay of the three TCFs follow
the order: (6V6V(t» > (6 Vzz 6Vzz (t» > (6Ez 6Ez (t» for
all solutes except S + . In all cases the Vzz and E z TCFs are
more similar to one another than they are to (6V6V(t».
With the exception of the small charged solutes S + ~ and
S +, the (6V6V(t» TCFs exhibit pronounced oscillatory
structure due to a collective librational motion of first shell
waters. Such oscillations are absent or greatly suppressed in
the (6Ez 6Ez (t» and (6Vz 6Vzz (t» TCFs.
(iii) Different solutes have different solvation times. All
of the large solutes show rather similar behavior, however, at
long times solvation of the uncharged (L 0) solute seems to
proceed more rapidly than'in the charged (L + and L - )
solutes. Differences in dynamics are much more pronounced
for the small solutes,' for which the rate of solvation decreases as the solute charge is increased in the series SO,
S +~,S +.
(iv) The solvation TCFs observed for the SO and L 0
solutes are indistinguishable to within our uncertainties.
Since these solutes differ in size by a factor of2.25, solute size
does not appear to be of primary importance in this range.
(v) The solvation TCFs decay much more rapidly than
any related single molecule TCFs. It is the intermolecular
correlations among molecules that cause rapid decay of
these equilibrium TCFs, or alternatively, it is the cooperativity of the solvation response that makes it so rapid. For
(6V6V(t» on the order of 10-40 molecules appear to be
sufficient to achieve the ultimate response time.
(vi) For the solutes studied, the first solvation shell
dominates the response, especially at short times. The relative contributions of librationaVreorientational vs vibrational/translational types of motions in the first shell response to a large extent determine the shape and average
5065
time constant of the various TCFs. Translational mechanisms of relaxation appear to increase in relative importance
as the range of the interaction becomes shorter and, for the S
solutes, as the charge is increased.
Comparison of the response to a charge jump measured
in nonequilibrium simulations to the corresponding TCF,
(6V6V(t», observed in equilibrium allow for the following
conclusions concerning the linearity of the solvation response.
(vii) Abrupt change of a full electronic charge in both
the L and the S solutes leads to a local heating of nearby
water molecules that dissipates relatively slowly on the solvation time scale. Even for the L solutes, for which local
heating offirst solvation shell waters involves a temperature
change of only - 20 K, this effect would be expected to significantly enhance the rate relative to the constant temperature case.
(viii) All charge jumps involving L solutes produce
very similar response functions, which also closely resemble
(6 V6 V(t) ) observed for LOin equilibrium. The nonequilibrium response appears to be slightly faster than the equilibrium TCF at long times. Thus a full charge jump in an L solute
can be considered "linear enough" for most purposes.
(ix) Jump of a full electronic charge in an S solute is
clearly nonlinear as witnessed by the very different relaxation times for the processes SO ..... S + and S + ..... S O. Even
for these cases, we note that predictions based on a linear
response calculation from the final state of the charge jump
provide estimates of the actual dynamics correct to within a
factor of2.
At this point it is worth mentioning the work of Karim
et al. 46 who recently performed nonequilibrium simulations
very similar to ours. These authors studied the response to
step function changes in dipole moment of a spherical solute
using the TIPS4p 30•76 model of water. Preliminary results of
this work indicate that changes of dipole moment of the order of 8 D may also lead to nonlinear behavior of the kind
observed here with unit charge jumps.
We now consider our simulation results in light of experimental measurements of solvation rates. Due to limitations of experimental time resolution it is appropriate to use
the long-time component of the simulated response for this
comparison. Based on the experimental correlation between
T.o1v / T Land £01£
observed in a variety of solvents21 •22 we
would anticipate the solvation response in ST2 water to decay with a time constant of - 9 ps (at long times). The simulated results are in clear disagreement with such an estimate. 77 None of the solutes studied exhibit such a long
component in any of their responses. Rather, our results
would indicate that water does not obey the same correlation
found with other solvents. At long times, the solvation energy relaxes with a time constant in the range -1-3 TL I ' or
-0.5-1.5 ps, for most solutes. Having made this comparison, it is worth pointing out that the present results are for
the ST2 water model of water and not for real water itself.
We cannot rule out the possibility that features of experimental solvents not included in this model, such as vibrational and electronic polarizability, may substantially alter
the time scales of solvation. For the present however, the
CL>
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5066
M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
similarity between the longitudinal relaxation times of the
ST2 model and real water encourages us to conjecture that
the solvation times observed here are representative of the
experimental solvent as well. Time-resolved Stokes shift
measurements of solvation in water have not yet been made
but are currently underway79 and should provide an important test of this prediction. We note that related measurements involving electron solvation in water have recently
been reported by Migus et al. 80 These workers observed the
absorption spectrum characteristic of the fully hydrated
electron to arise with a time constant of 0.24 ps. This value,
while in reasonable agreement with the times observed here,
must be viewed with caution. Changes in the solvated electron spectrum seem to reflect a change in electronic state
during solvation rather than simply a continuous relaxation
of the solvent surroundings80 as in our simulations or in experiments with other probes. Interpretation of the electron
solvation time is thus uncertain.
Finally, we compare the behavior observed in our simulations to current theoretical ideas of the solvation response.
In Sec. VI we showed that neither the simple continuum
model nor the best presently available molecular model can
account quantitatively for the simulated results. Here we
want to consider how well the qualitative features of our data
compare to expectations. It is generally accepted that the
solvation response time predicted from simple continuum
models (i.e., 7'L in Debye solvents) should provide a lower
limit to the actual solvation time scale. If we ignore the
S + -+ S 0 case, we do not observe response times significantly faster than the continuum prediction. In many cases, however, the times we do observe are very close to the continuum
prediction, and are much faster than predicted by the molecular, MSA theory. The reason for the disagreement appears
to be due to the fast response contributed by first solvation
shell waters. As discussed in Sec. I, deviations from simple
continuum behavior are usually thought of as resulting from
a variation of the response times of the solvent with distance
from the solute. Solvent nearest the solute is pictured as having a relatively slow response, and only far from the solvent
does the limiting, continuum time scale pertain. We do observe that correlation functions of the radial shell polarizations [Fig. 14(B)] decay most slowly near to the solute.
However, the potential TCF of the first shell is actually faster than that ofthe other shells [Fig. 14(A)], which decay
similarly to their respective polarization TCFs. Thus the solvation response, measured by (c5Vc5V(t», is sensitive to factors not taken into account by the shell polarization alone.
The fact that the solute senses the full charge distribution of
nearby solvent molecules, rather than seeing them as simply
point dipoles, changes the dynamics, in this case making it
faster. 8 I
The importance of the first shell response as well as its
complex character is perhaps the most important result of
our study. Effects due to the extended charge distribution of
the solvent, and translational modes of relaxation become
important at close range. Such effects are not represented in
the measured €(w), nor are they included in any presently
available molecular model. Not only do these effects change
the quantitative result, they also cause disagreement with
the qualitative picture of a slow nearest-neighbor response.
All current theories of solvation dynamics assume that
the solvent responds linearly to changes in the solute. In Sec.
IV we observed that the eqUilibrium solvation energies were
linear with respect to changes in solute charge over a wide
range of charge/size ratios. With respect to dynamics, we
observe that although a unit charge jump cannot strictly be
described by a linear response, in most cases the linear approximation provides a reasonable representation of the actual response.
We conclude by observing that for a theoretical model
to adequately describe the dynamics of solvation in water, a
much more sophisticated treatment of the solvation structure seems necessary. Nonlinear effects, while present, are
less important than is properly accounting for the detailed
nature of the first solvation shell interactions which lead to
the dynamic differences observed with different solutes and
responses. A model sufficient to accurately predict the effects we observe would likely be too complex to be instructive. Water is one of the most highly structured of solvents
and so its dynamics are probably more complex and solute
specific than other polar solvents. However, the importance
of the extended charge distribution of solvent molecules and
translational mechanisms of relaxation observed for water
should pertain to other solvents as well. Thus, development
of simple theoretical models capable of treating such effects
would be of interest. So too would further simulations using
simpler model solvents. We are currently beginning such
simulations with a model acetonitrile solvent.
ACKNOWLEDGMENTS
The work reported here was supported by grants from
the National Science Foundation. We also acknowledge generous allocations of computer time at the National Center
for Supercomputing Applications under Grant No.
CHE8303997. During the course of this work we have benefited from numerous discussions with Peter Wolynes, Casey
Hynes, Paul Barbara, John Simpn, Biman Bagchi, and Ed
Castner. We would also like to thank Stuart Rice and Michael Townsend providing us computer code for the ST2
interaction potential, and Martin Neumann for sharing results of his determination of E(W) ofST2 water prior to publication.
APPENDIX: ESTIMATION OF THE DIELECTRIC
PROPERTIES OF ST2 WATER
In order to compare our solvation dynamics results to
theoretical predictions, the dielectric dispersion E( OJ) of the
solvent is needed as input. Since this was not known for the
ST2 model we made a preliminary determination based on
pure cluster simulations. Neumann61 is now in the process of
making a more accurate determination of €(w) and although we employ his results in Sec. VI, we present our cluster calculations here for the sake of comparison.
The method we employ is due to Bossis and coworkers. 82 •83 The static dielectric constant is obtained via
simulation of a generalized, radially dependent Kirkwood g
factor G( r/R) for a spherical sample of radius R. G( r/R) is
defined by
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M. Maroncelli and G. R. Flaming: Dynamics of aqueous solvation
2
G(rIR) = ( M (r») ,
p?n(r)
(A1)
where n (r) and M (r) are the number of molecules and total
moment of a (centered) spherical region of radius rand /.l is
the molecular dipole moment. This quantity is related to the
dielectric constant Eo by82,83:
G(rIR)=~ (Eo-I)
y 9Eo(Eo + 2)
X { (Eo + 2)( 2Eo + 1) - 2(Eo - 1)2 (
~r}
(A2)
with
_ 41T'pp?
y- 9kT .
Relations (A 1) and (A2) are derived for macroscopic samples. In order to apply the relation to microscopic clusters
Eq. (AI) must be replaced by82:
G{r/R)
=
M(r - rc )-M(r) )
2
( /.In(r-rc)
5067
(rlR)3 dependence except near to the cluster boundary. (In
this region the error bars are approximately the size of the
data points.) Based on the range of the statistical error alone,
a value of Eo = 340 ± 90 is obtained. The actual error is likely to be greater than this due to the relatively small size of the
cluster used. This makes the calculation sensitive to the
choices of R and rc as well as possible effects due to preferential surface orientation of molecules. For these reasons we
estimate the static dielectric constant of ST2 water to be
340 ± 150 at 291 K. Even with such wide error limits it is
clear that the ST2 model grossly overestimates the dielectric
constant of water.
The frequency dependence of the dielectric response can
be estimated from the same simulation from the time correlation function of the total dipole moment of the cluster
(M-M(t». E(W) is related to the normalized TCF ~(t),
cI»(t)
= (M-M(t»/(M 2 )
(A4)
by
(E(W)-l)( Eo-l
(A3)
'
where rc is a length on the order of a few molecular diameters
over which dipolar orientations are strongly correlated.
Plotting G(rIR) obtained from simulated M(r) data according to Eq. (A3) vs (rIR)3 should yield a straight line
whose slope gives Eo.
Figure 21 shows the results of such a calculation of Eo
using a simulation of a cluster of 512 molecules at 291 K. The
simulation was run for a total of 250 ps using a step size of 5
fs. The cluster radius R was taken to be 4.970', which is the
radius the sample would have if the density profile were a
step function. The correlation length rc was chosen to be 20'.
Error bars in this figure reflect the statistical uncertainty due
to limited simulation length. The solid lines are the behavior
expected based on Eq. (A2) for two values of Eo that bracket
the statistical error range. To within the statistical uncertainties, the simulated data are adequately represented by an
{_dcl»},
(A5)
dt
where .!f iOJ denotes a Laplace transform. Figure 22 shows
the cI»( t) obtained from the simulation described above. The
error bar shown on the graph is again the estimated statistical uncertainty in the averaging. cI» ( t) can be represented by
a sum of two exponentials as
Eo+2)=.!fiw
E(w)+2
cI»(t)
=/e- 1IT , + (l-/)e- 1IT,
(A6)
with values /= 0.72,1'1 = 0.76 ps, and 1'2-0.012 ps. This
result is independent of the value of Eo. We obtained parameters describing the E(W) function by performing the Laplace transform in Eq. (A5) on this fitted «I>(t). The resultant dielectric dispersion can be represented by a sum of two
Debye processes with the constants
EOl = Eo = 340,
Eool =E02 =2.1,
E002
= Eoo
1'DI
= 64 ps,
1'D2
=0.016ps,
= 1.
It is instructive to compare these cluster results to those
obtained by Neumann61 using a periodic simulation. (The
later results are provided in Sec. VI.) The dielectric constant
12
9
0
([
.........
'-
6
A
t!)
-1
+oJ
3
~
*
-2
~
v
0
0
0.5
1
(r /R)3
FIG. 21. Generalized Kirkwood gfactor G(r/R); Eq. (A3) observed in a
250 ps simulation of a pure 512 cluster. The error bars show the statistical
uncertainty due to finite simulation length. The solid lines show the range of
linear fits to Eq. (A2) consistent with this uncertainty. The two lines correspond to Eo = 253 (upper) and Eo = 432 (lower).
c -3
--l
-4
0
1
Time
2
(ps)
3
FIG. 22. Correlation function of the total dipole moment [Eq. (A4) 1ofa
pure 512 cluster.
J. Cham. Phys., Vol. 89, No.8, 15 October 1988
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5068
M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
that we obtain is considerably higher than Neumann's value
of218, however, they are consistent to within the estimated
error limits. The values of T D 1 differ by a factor of 2 between
the two determinations but this is simply a reflection of the
difference in Eo. The long-time part of <I> (t) decays on a time
scale that is approximately proportional to T D 1 / Eo. Error in
Eo then appears as a proportional error in T D 1 in cluster simulations. (This is not true of bulk simulations.) Rather than
comparing the E(W) parameters derived by the two simulations, the more relevant comparison as far as solvation dynamics is concerned is to compare the time dependence of
<I> (t). <I> (t) decays in a manner closely related to the decay of
longitudinal dielectric fluctuations which, in the continuum
limit, determine solvation dynamics. Using the inverse of
Eq. (A5), Neumann's E( w) results can be transformed into
a <1>(1) to compare to the function measured directly here.
Treating Neumann's E(W) in this manner yields a <1>(1) decay that agrees with our result to well within our estimated
error. This agreement between our results and the bulk simulations of Neumann is noteworthy. It indicates that the
presence ofthe cluster surface, while it may influence determination of Eo and E(W), should not appreciably affect the
solvation dynamics of most interest to us here.
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I
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helpful in understanding the origin of artifacts observed in early simulations.
5lIt is worth pointing out that the use of periodic boundary conditions does
not eliminate the need for large sample sizes. For example, in the Ewald
summation method the long-range interactions between molecules in the
central simulation cell and molecules in an infinite periodic array of replica cells are accounted for exactly via a lattice summation. If one is concerned about solvation of a single ion, the periodicity imposes artifacts
near the cell boundaries analogous to those in a cluster. Thus, as one
crosses the boundary from the central cell to a replica cell, the polarization
of the solvent reverses sign in an unphysical way. The error incurred via
this effect may in fact be worse than that obtained using free clusters. Only
with sufficiently large samples does this effect become negligible.
52L. Yerlet, Phys. Rev. 159,98 (1967).
S3J._P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J. Comput. Phys. 23,
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54H. C. Andersen, J. Comput. Phys. 52, 24 (1983).
55R. M. Townsend, J. Gryko, and S. A. Rice, J. Chern. Phys. 82, 4391
(1985).
56J. G. Powles, J. Chern. Phys. 21, 633 (1953).
57U. Kaatze and Y. Uhlendorf, Z. Phys. Chern. N. F. 126, lSI (1981).
s8M. R. Mruzik, Chern. Phys. Lett. 48, 171 (1977).
59E. N. Brodskaya and A.I. Rusanov, Mol. Phys. 62, 251 (1987).
bOA. P. Shreve, J. P. R. B. Walton, and K. E. Gubbins, J. Chern. Phys. 85,
2178 (1986).
61M. Neumann (private communication).
62D. Y. C. Chan, D. J. Mitchell, and B. W. Ninham, J. Chern. Phys. 70, 2946
(1979).
63See, for example, C. G. Gray and K. E. Gubbins, Theory of Molecular
J. Chern. Phys., Vol. 89, No.8, 15 October 1988
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M. Maroncelli and G. R. Fleming: Dynamics of aqueous solvation
Fluids (Oxford, New York, 1984), Vol. I, Chap. S.
MG. Engel and H. G. Hertz, Ber. Bunsenges. Phys. Chem. 71, 808 (1968).
MExperimentally the librational motion in liquid water gives rise to a broad
band at - 600 cm - I in the IR spectrum (Ref. 66), in approximate agreement with the time scale observed for the ST2 model.
~, for example, D. Eisenberg and W. Kauzmann, The Structure and
Properties of Water (Oxford, New York, 1969), Sect. 4.7.
67 Apparently similar librational oscillations were observed in the decay of
the longitudinal dipole moment correlation function ofMCY water by R.
W. lmpey, P. A. Madden, and I. R. McDonald, Mol. Phys. 46, 513
(1982).
68-J'he force autocorrelation functions determined in Ref. 4 and especially in
Refs. 42 and 44 are closely related to (liE.liE. (t» TCFs determined here.
Unfortunately, these force autocorrelation functions include forces due to
van der Waals interactions in addition to electrostatic forces and so are not
directly comparable.
69
0. Matsuoka, E. Clementi, and M. Yoshimine, J. Chem. Phys. 64,1351
(1976); G. C. Lie, E. Clementi, and Y. Yoshimine, ibid. 64, 2314 (1976).
7<lM. Neumann, J. Chem. Phys. 81, 5663 (198S).
7IActually, a more appropriate comparison to the experimental TL is with
the long-time decay of the continuum prediction for Set) of an ion [Fig.
12(B)]. This time constant is 0.49 ps, slightly longer than TLI due to the
effect of the second dispersion region.
72Nichols and Calef (Ref. 28) have, however, obtained a solution for arbitrary solute size ofa single Debye E( I i ) .
5069
73We have calculated the single-molecule contribution to (liVliV(t» in a
few cases but do not show the results here because of poor signal to noise.
7"These two types of intramolecuiar motions have vibrational character at
short times and are what give rise to librational and hindered translational
modes observed in the IR and Raman spectra of liquid water (Ref. 66).
"J. Jonas, T. DeFries, and D. J. Wilbur, J. Chern. Phys. 65, S82 (1976).
7~. L. Jorgensen and J. D. Madura, Mol. Phys. 56,1381 (1985).
77In a preliminary communication of the work presented here (Ref. 78), we
concluded that the simulation results were consistent with the experimental estimate. This erroneous conclusion was due to poor estimates of the
longitudinal relaxation time and dielectric constant ofST2 water.
78M. Maroncelli, E. W. Castner, Jr., S. P. Webb, and G. R. Fleming, in
Ultrafast Phenomena V, edited by G. R. Fleming and A. E. Siegman
(Springer, Heidelberg, 1986), p. 303.
79A. Ruggiero, S. J. Rosenthal, L. Dhar, and G. R. Fleming (in preparation).
BOA. Migus, Y. Gaudel, J. L. Martin, and A. Antonetti, Phys. Rev. Lett. 58,
1559 (1987).
81 Although not shown, we see the same differences between the shell polarizations and the potentials in the nonequilibrium simulations that we do in
the equilibrium case.
82C. Hesse-Bezot, G. Bossis, and C. Brot, J. Chem. Phys. 80,3399 (1984);
G. Bossis, Mol. Phys. 38, 2023 (1979).
83See also J. G. Powles and R. F. Fowler, J. Phys. C 18,5909 (1985); J. G.
Powles, R. F. Fowler, and W. A. B. Evans, Chern. Phys. Lett. 107,280
(1984).
J. Chern. Phys., Vol. 89, No.8, 15 October 1988
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